Wikiversity:Notices for custodians

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== Notice about granting curator rights ==

As of now, only bureaucrats can grant and remove curator rights because curator has some custodian-level permissions (delete [but not undelete], protect, etc.). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:19, 7 July 2026 (UTC)

== Kids train land channel  ==

Says Ok  As of now, only bureaucrats can grant and remove curator rights because curator has some custodian-level permissions (delete [but not undelete], protect, etc.). [[Special:Contributions/&amp;#126;2026-47369-73|&amp;#126;2026-47369-73]] ([[User talk:&amp;#126;2026-47369-73|talk]]) 21:08, 30 August 2026 (UTC)</text>
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      <comment>Reverted edit by [[Special:Contributions/~2026-47369-73|~2026-47369-73]] ([[User_talk:~2026-47369-73|talk]]) to last version by [[User:Codename Noreste|Codename Noreste]] using [[Wikiversity:Rollback|rollback]]</comment>
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== Notice about granting curator rights ==

As of now, only bureaucrats can grant and remove curator rights because curator has some custodian-level permissions (delete [but not undelete], protect, etc.). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:19, 7 July 2026 (UTC)</text>
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  <page>
    <title>Social problems</title>
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      <text bytes="3295" sha1="4g0d4pcbj7mqk41krn9r1vv06ft6fo9" xml:space="preserve">The following are learning resources related to '''social problems'''. Attempting to work towards [[solving problems]] in living can be a to try and reduce the undesirable impact of social problems. 
__NOTOC__
== What are social problems? ==
Social problems are the general factors that affect and damage society.  Also known as social issues sometimes. A social problem is normally a term used to describe problems with a particular area or group of people in the world. Social problems often involve problems that affect the real world. It also affects how people react to certain situations. Examples can include:
{{Col}}
* Illiteracy 
* [[wikipedia:Anti-social_behaviour|Anti social behavior]]/crime
** Rape
** Crime and [[Digital Media Concepts/Cyber Crimes|Cyber crime]]
** [[wikipedia:Human_trafficking|Human Trafficking]]
** Sexual abuse
** High crime rate
** [[Child abuse]]
{{ColBreak}}
* [[wikipedia:Malnutrition|Malnutrition]]
*Discrimination
** Racial discrimination
** Ageism
** Sexism
** Ableism
* The shortage of schools
* Political corruption
* Unemployment/underemployment
* Unplanned/early pregnancy
{{ColBreak}}
* [[wikipedia:Poverty|Poverty]]
* [[wikipedia:Substance_abuse|Drug abuse]]
*[[wikipedia:Alcohol_abuse|Alcohol abuse]]
* Bullying
* Obesity
* Irresponsible/problematic gambling
* Animal abuse
* The lack of infrastructure
{{ColBreak}}
* Economic deprivation
** Economic injustices
* [[wikipedia:Female_genital_mutilation|Female genital mutilation]]
* War
* Unhappiness/misery/lack of meaning
** Stress (distress, not eustress)
** Lack of social skills (which some may not feel is a problem)
* Family conflict/family disharmony

{{Col/end}}

== Discussion questions and essay ideas ==
* What are some of the most prevalent social problems that exist? 
* What are some examples of &quot;problems in living&quot;, and how does these relate to social problems? 
* How can people work together to remedy social problems?
* Is lack of education a social problem? How can embracing a voluntary education be encouraged?
* How can open source content on wikis help to develop solutions to social problems?
* What are the most efficient ways to help eliminate poverty on Earth?
* Is paying drug abusers to be sober a sustainable and effective tactic?

==Readings==
===Wikipedia===
* [[w:Nuclear labor issues|Nuclear labor issues]] - Labor issues can be social problems. Labor issues can relate to economic problems which can affect individuals and families. 
* [[w:List of ongoing armed conflicts|List of ongoing armed conflicts]]
* [[w:List_of_proxy_wars#Ongoing_proxy_wars|List of ongoing proxy wars]]

== See also ==
* [[Grand Challenges]]
*[[Exploring Social Constructs]]
* [[Eliminating poverty]]
* [[Counseling psychology]]
* [[Wikidebate]]
* [[Social environment]]
* [[Economic problems]]
* [[Spiritual problems]]
* [[Ecological problems]]
* [[Political problems]]
* [[Health problems]]
* [[Philosophical problems]]
* [[Technological problems]]
* [[Environmental problems]] - Social problems can contribute to environmental problems, and environmental problems can interact with or worsen social problems.

== External links ==
* [http://freerice.com/ Freerice.com]

[[Category:Social psychology]]
[[Category:Social research]]
[[Category:Sociology]]
[[Category:Social problems]]</text>
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      <text bytes="3371" sha1="kwr3mg6qnm1vjhrk6l5936ptd71pgjb" xml:space="preserve">The following are learning resources related to '''social problems'''. Attempting to work towards [[solving problems]] in living can be a to try and reduce the undesirable impact of social problems. 
__NOTOC__
== What are social problems? ==
Social problems are the general factors that affect and damage society.  Also known as social issues sometimes. A social problem is normally a term used to describe problems with a particular area or group of people in the world. Social problems often involve problems that affect the real world. It also affects how people react to certain situations. Examples can include:
{{Col}}
* Illiteracy 
* [[wikipedia:Anti-social_behaviour|Anti social behavior]]/crime
** Rape
** Crime and [[Digital Media Concepts/Cyber Crimes|Cyber crime]]
** [[wikipedia:Human_trafficking|Human Trafficking]]
** Sexual abuse
** High crime rate
** [[Child abuse]]
{{ColBreak}}
* [[wikipedia:Malnutrition|Malnutrition]]
*Discrimination
** Racial discrimination
** Ageism
** Sexism
** Ableism
* The shortage of schools
* Political corruption
* Unemployment/underemployment
* Unplanned/early pregnancy
{{ColBreak}}
* [[wikipedia:Poverty|Poverty]]
* [[wikipedia:Substance_abuse|Drug abuse]]
*[[wikipedia:Alcohol_abuse|Alcohol abuse]]
* Bullying
* Obesity
* Irresponsible/problematic gambling
* Animal abuse
* The lack of infrastructure
{{ColBreak}}
* Economic deprivation
** Economic injustices
* [[wikipedia:Female_genital_mutilation|Female genital mutilation]]
* War
* Unhappiness/misery/lack of meaning
** Stress (distress, not eustress)
** Lack of social skills (which some may not feel is a problem)
* Family conflict/family disharmony

{{Col/end}}

== Discussion questions and essay ideas ==
* What are some of the most prevalent social problems that exist? 
* What are some examples of &quot;problems in living&quot;, and how does these relate to social problems? 
* How can people work together to remedy social problems?
* Is lack of education a social problem? How can embracing a voluntary education be encouraged?
* How can open source content on wikis help to develop solutions to social problems?
* What are the most efficient ways to help eliminate poverty on Earth?
* Is paying drug abusers to be sober a sustainable and effective tactic?

==Readings==
===Wikipedia===
* [[w:Nuclear labor issues|Nuclear labor issues]] - Labor issues can be social problems. Labor issues can relate to economic problems which can affect individuals and families. 
* [[w:List of ongoing armed conflicts|List of ongoing armed conflicts]]
* [[w:List_of_proxy_wars#Ongoing_proxy_wars|List of ongoing proxy wars]]

== See also ==
* [[Grand Challenges]]
*[[Exploring Social Constructs]]
* [[Eliminating poverty]]
* [[Counseling psychology]]
* [[Wikidebate]]
* [[Social environment]]
* [[Economic problems]]
* [[Spiritual problems]]
* [[Ecological problems]]
* [[Legal problems]]
* [[Emotional problems]]
* [[Infrastructure problems]]
* [[Political problems]]
* [[Health problems]]
* [[Philosophical problems]]
* [[Technological problems]]
* [[Environmental problems]] - Social problems can contribute to environmental problems, and environmental problems can interact with or worsen social problems.

== External links ==
* [http://freerice.com/ Freerice.com]

[[Category:Social psychology]]
[[Category:Social research]]
[[Category:Sociology]]
[[Category:Social problems]]</text>
      <sha1>kwr3mg6qnm1vjhrk6l5936ptd71pgjb</sha1>
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      <text bytes="3396" sha1="86ymyojcsclorzoaohz0p4klrj6ph0m" xml:space="preserve">The following are learning resources related to '''social problems'''. Attempting to work towards [[solving problems]] in living can be a to try and reduce the undesirable impact of social problems. 
__NOTOC__
== What are social problems? ==
Social problems are the general factors that affect and damage society.  Also known as social issues sometimes. A social problem is normally a term used to describe problems with a particular area or group of people in the world. Social problems often involve problems that affect the real world. It also affects how people react to certain situations. Examples can include:
{{Col}}
* Illiteracy 
* [[wikipedia:Anti-social_behaviour|Anti social behavior]]/crime
** Rape
** Crime and [[Digital Media Concepts/Cyber Crimes|Cyber crime]]
** [[wikipedia:Human_trafficking|Human Trafficking]]
** Sexual abuse
** High crime rate
** [[Child abuse]]
{{ColBreak}}
* [[wikipedia:Malnutrition|Malnutrition]]
*Discrimination
** Racial discrimination
** Ageism
** Sexism
** Ableism
* The shortage of schools
* Political corruption
* Unemployment/underemployment
* Unplanned/early pregnancy
{{ColBreak}}
* [[wikipedia:Poverty|Poverty]]
* [[wikipedia:Substance_abuse|Drug abuse]]
*[[wikipedia:Alcohol_abuse|Alcohol abuse]]
* Bullying
* Obesity
* Irresponsible/problematic gambling
* Animal abuse
* The lack of infrastructure
{{ColBreak}}
* Economic deprivation
** Economic injustices
* [[wikipedia:Female_genital_mutilation|Female genital mutilation]]
* War
* Unhappiness/misery/lack of meaning
** Stress (distress, not eustress)
** Lack of social skills (which some may not feel is a problem)
* Family conflict/family disharmony

{{Col/end}}

== Discussion questions and essay ideas ==
* What are some of the most prevalent social problems that exist? 
* What are some examples of &quot;problems in living&quot;, and how does these relate to social problems? 
* How can people work together to remedy social problems?
* Is lack of education a social problem? How can embracing a voluntary education be encouraged?
* How can open source content on wikis help to develop solutions to social problems?
* What are the most efficient ways to help eliminate poverty on Earth?
* Is paying drug abusers to be sober a sustainable and effective tactic?

==Readings==
===Wikipedia===
* [[w:Nuclear labor issues|Nuclear labor issues]] - Labor issues can be social problems. Labor issues can relate to economic problems which can affect individuals and families. 
* [[w:List of ongoing armed conflicts|List of ongoing armed conflicts]]
* [[w:List_of_proxy_wars#Ongoing_proxy_wars|List of ongoing proxy wars]]

== See also ==
* [[Grand Challenges]]
*[[Exploring Social Constructs]]
* [[Eliminating poverty]]
* [[Counseling psychology]]
* [[Wikidebate]]
* [[Social environment]]
* [[Economic problems]]
* [[Spiritual problems]]
* [[Ecological problems]]
* [[Legal problems]]
* [[Emotional problems]]
* [[Problems in living]]
* [[Infrastructure problems]]
* [[Political problems]]
* [[Health problems]]
* [[Philosophical problems]]
* [[Technological problems]]
* [[Environmental problems]] - Social problems can contribute to environmental problems, and environmental problems can interact with or worsen social problems.

== External links ==
* [http://freerice.com/ Freerice.com]

[[Category:Social psychology]]
[[Category:Social research]]
[[Category:Sociology]]
[[Category:Social problems]]</text>
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    <title>Wikiversity talk:Why create an account</title>
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      <comment>/* Kids train land channel 🤔💭🚂 🌍 🙏 */ new section</comment>
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      <text bytes="1965" sha1="n5or806s8e3qg3rwulm72kt61xrw6as" xml:space="preserve">Added a frame and a caption to the screenshot for better usability, as I noticed that it was not obviously an image when the article is browsed quickly. I got confused at first, until I read the text above it. Couldn't get the image width to work for some reason, needs to be fixed.


==remark: explain that it's global right at begin==
was logged out and got presented: &quot;Why create a Wikiversity account? Not logged in. Please log in to view or edit items on your watchlist.&quot; I was wondering why restrict this account to WV only? when there is right there told it is a global account, it could be more activating. Anyways, later... ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] &lt;small&gt;uses the [[Wikiversity:Chat]] + [[Identi.ca]]&lt;/small&gt; 01:31, 9 May 2011 (UTC)
大FAS

== Wikiversity ==

Wikiversity is a Wikimedia Foundation project that supports learning communities, their learning materials, and resulting activities. It differs from Wikipedia in that it offers tutorials and other materials for the fostering of learning, rather than an encyclopedia; like Wikipedia, it is available in many languages. [[User:Butterfly Florist|Butterfly Florist]] ([[User talk:Butterfly Florist|discuss]] • [[Special:Contributions/Butterfly Florist|contribs]]) 06:26, 20 September 2022 (UTC)

It offers tutorials  [[User:Butterfly Florist|Butterfly Florist]] ([[User talk:Butterfly Florist|discuss]] • [[Special:Contributions/Butterfly Florist|contribs]]) 07:47, 27 September 2022 (UTC)

== The importance of having a wikiversity account ==

Nothing [[User:Butterfly Florist|Butterfly Florist]] ([[User talk:Butterfly Florist|discuss]] • [[Special:Contributions/Butterfly Florist|contribs]]) 06:47, 27 September 2022 (UTC)

== Kids train land channel 🤔💭🚂 🌍 🙏 ==

🤔💭😁😁Kids train land channel 🤔💭🚂 🌍 🙏 [[Special:Contributions/&amp;#126;2026-47369-73|&amp;#126;2026-47369-73]] ([[User talk:&amp;#126;2026-47369-73|talk]]) 21:06, 30 August 2026 (UTC)</text>
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        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>Reverted edit by [[Special:Contributions/~2026-47369-73|~2026-47369-73]] ([[User_talk:~2026-47369-73|talk]]) to last version by [[User:Atcovi|Atcovi]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <text bytes="1712" sha1="7uiuyaaluiur117wwf5xwovq1b0tmxs" xml:space="preserve">Added a frame and a caption to the screenshot for better usability, as I noticed that it was not obviously an image when the article is browsed quickly. I got confused at first, until I read the text above it. Couldn't get the image width to work for some reason, needs to be fixed.


==remark: explain that it's global right at begin==
was logged out and got presented: &quot;Why create a Wikiversity account? Not logged in. Please log in to view or edit items on your watchlist.&quot; I was wondering why restrict this account to WV only? when there is right there told it is a global account, it could be more activating. Anyways, later... ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] &lt;small&gt;uses the [[Wikiversity:Chat]] + [[Identi.ca]]&lt;/small&gt; 01:31, 9 May 2011 (UTC)
大FAS

== Wikiversity ==

Wikiversity is a Wikimedia Foundation project that supports learning communities, their learning materials, and resulting activities. It differs from Wikipedia in that it offers tutorials and other materials for the fostering of learning, rather than an encyclopedia; like Wikipedia, it is available in many languages. [[User:Butterfly Florist|Butterfly Florist]] ([[User talk:Butterfly Florist|discuss]] • [[Special:Contributions/Butterfly Florist|contribs]]) 06:26, 20 September 2022 (UTC)

It offers tutorials  [[User:Butterfly Florist|Butterfly Florist]] ([[User talk:Butterfly Florist|discuss]] • [[Special:Contributions/Butterfly Florist|contribs]]) 07:47, 27 September 2022 (UTC)

== The importance of having a wikiversity account ==

Nothing [[User:Butterfly Florist|Butterfly Florist]] ([[User talk:Butterfly Florist|discuss]] • [[Special:Contributions/Butterfly Florist|contribs]]) 06:47, 27 September 2022 (UTC)</text>
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  <page>
    <title>Wikiversity talk:Learning resources</title>
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      <id>2829768</id>
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      <contributor>
        <username>~2026-47369-73</username>
        <id>3110319</id>
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      <comment>/* Kids train land channel 🤔💭 👍👍 🚂 🌍 😁 */ new section</comment>
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      <text bytes="470" sha1="6p639cwkslegvf3uhec5cja33brun5s" xml:space="preserve">[[Wikiversity:Learning resources|This page]] now has its original purpose. Older discussion of another topic was moved to another page; see [[Wikiversity talk:Being educational]]. --[[User:JWSchmidt|JWSchmidt]] 14:56, 17 June 2010 (UTC)

== Kids train land channel 🤔💭 👍👍 🚂 🌍 😁 ==

Kids train land channel of world  [[Special:Contributions/&amp;#126;2026-47369-73|&amp;#126;2026-47369-73]] ([[User talk:&amp;#126;2026-47369-73|talk]]) 19:54, 30 August 2026 (UTC)</text>
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      <minor />
      <comment>Reverted edit by [[Special:Contributions/~2026-47369-73|~2026-47369-73]] ([[User_talk:~2026-47369-73|talk]]) to last version by [[User:JWSchmidt|JWSchmidt]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <text bytes="236" sha1="micy4kole0nksd27fqb6iba8txo3u4d" xml:space="preserve">[[Wikiversity:Learning resources|This page]] now has its original purpose. Older discussion of another topic was moved to another page; see [[Wikiversity talk:Being educational]]. --[[User:JWSchmidt|JWSchmidt]] 14:56, 17 June 2010 (UTC)</text>
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      <comment>returned from international travel</comment>
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{{userboxtop}}
{{User Male}}
{{User Muslim}}
{{Sri Lankan}}
{{User contrib|25000}}
{{User SUL Box|2=v}}
{{User Wikiversitan For|year=2011|month=1|day=28}}
{{User custodian}}
{{User admin Wikibooks}}
{{User admin MediaWiki}}
{{Global rollbacker}}
{{User Meta-Wiki}}
{{User researcher}}
{{User soccer}}
{{User psychology}}
{{userboxbottom}}{{Introduction}}
I'm Atcovi, a member of the English Wikiversity community since January 2011. I currently serve as a [[Wikiversity:Custodianship|custodian]] (since June 2021) and a [[Wikiversity:Bureaucratship|bureaucrat]] (since May 2026). My academic interests mainly lie within [[School:Psychology|psychology]], specifically [[clinical psychology]] (with subfields of interest being [[suicidology|suicide]] and [[General Psychopathology|psychopathology]]).  

My activity is high at the moment but may fluctuate due to life circumstances. Reach out to my talk page for any inquiries. 

===Links===
[[File:Sura Minshawi 2.ogg|thumb|left|[[w:Muhammad_Siddiq_Al-Minshawi|Sheikh Minshawi's]] recitation of Surah Al-Baqara]]
[[File:Notifications-Talk-Indicator-OptionG-OBOD -Screenshot-Closeup-05-01-2013.png|thumb|right|I remember when I used to get these notifications... (2013)]]
I've left an arrangement of random links for me to easily access if I so desire at any given time.  
# [[Help:Project boxes]] - For projects/pages.
# [[Help:Quiz]] - This is also important. 
# [https://en.wikiversity.org/w/index.php?title=Special%3APrefixIndex&amp;prefix=User%3AAtcovi%2F&amp;namespace=0 Pages under &quot;User:Atcovi&quot;] 
# [[Special:CentralAuth/Atcovi]]
# [[:Category:Atcovi's Work]], [[User:Atcovi/Science]] &amp; [[User:Atcovi/History]]
# https://tools.wmflabs.org/meta/crossactivity/Atcovi
# https://tools.wmflabs.org/topviews/?project=en.wikiversity.org&amp;platform=all-access&amp;date=yesterday&amp;excludes=
# &lt;code&gt;&lt;nowiki&gt;{{under construction}}&lt;/nowiki&gt;&lt;/code&gt;
#[https://en.wikipedia.org/wiki/Category:Psychology_stubs Psychology stubs] and [https://en.wikipedia.org/wiki/Category:Health_stubs Health stubs]
#[[User:Atcovi/Essays]]
#[[:Category:Featured resources]]
#[[Special:BrokenRedirects]]
#[[:Category:Stub templates]]
#[[User:Atcovi/To merge]]
{{Languages and skills|en-N|de-2}}
{{User:Atcovi/to do}}
== Wikiversity's To-do ==
{{Opentask}}
[[File:Flagge Palaestina.jpg|350px|frameless|center]]
[[Category:Wikiversity custodians]]
[[Category:Wikiversity bureaucrats]]
[[Category:User pages]]
[[Category:Atcovi's Work]]</text>
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  <page>
    <title>User:Michael Ten</title>
    <ns>2</ns>
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    <revision>
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      <timestamp>2026-08-30T18:32:02Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
      </contributor>
      <comment>/* More */  User:Michael Ten/common.css</comment>
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      <text bytes="4914" sha1="tstbqw0rf4hphuofdfmeseny8owipsa" xml:space="preserve">Michael Ten is a pen name of mine. 
&lt;!-- If you believe overcoming pro-aging mindsets is important please [http://patreon.com/defeataging support me on Patreon]. --&gt;
Please help me to help people to overcome pro-aging mindsets. 
* [http://michaelten.net Ideas]
__NOTOC__
What if we utilize [[vertical farming]] to convert farmland back into forests? What if we use [[electric cars]] and [[green energy]] to drastically reduce pollution in cities and all over? Some universities study ideas like [[basic income]] that [https://basicincome.stanford.edu/about/what-is-ubi/ may help] to eliminate poverty. [[Automation]] is increasingly changing economic dynamics on Earth. 
* [https://wikipedialibrary.wmflabs.org/users/my_library/ Library]
==Miscellaneous==
{{colbegin|4}}
* [https://en.wikiversity.org/wiki/Special:RecentChanges?hidebots=1&amp;hidecategorization=1&amp;hideWikibase=1&amp;namespace=0&amp;limit=1000&amp;days=15&amp;enhanced=1&amp;urlversion=2 Resources]
* [[Strategies for Engineered Negligible Senescence]]
* [[Overcoming pro-aging mindsets]]
* [[Android programming]]
* [[Renewable energy]]
* [[Ologies]]
* [[Pro se legal representation]]
* [[Business/Employee-owned companies|Employee-owned companies]]
* [[WikiJournal of Business and Economics]]
* [[Technological automation]]
* [[Online social entrepreneurship]]
* [[Large language models]]
* [[Social entrepreneurship]]
* [[Child abuse]]
* [[Business/Earning money]]
* [[Basic income cryptocurrency]]
* [[Open education]]
* [[Affordable housing]]
* [[Careers and Employment/Types of employment|Types of employment|]]
* [[Earthquake-resistant structures]]
* [[Business]]
* [[Basic income]]
* [[Marketing/Marketing educational wikis|Marketing educational wikis]]
* [[Szaszian studies]]
* [[Szaszian]]
* [[Szaszian theory]]
* [[Anti-psychiatry]]
* [[Audacity/Processing vocals|Processing vocals]]
* [[Should suicide be legal?‎‎]]
* [[Social problems]]
* [[Vertical farming]]
* [[Environmental problems]]
* [[Energy storage]]
* [[Eliminating poverty]]
* [[Futurism]]
* [[Nonprofit management]]
* [[Reddit for learning]]
* [[Learning by failing]]
* [[Cryonics]]
* [[Rhyming words for songwriters]]
* [[Product development]]
* [[Music production]]
* [[Online Industrial Community]] 
* [[User:Michael_Ten/common.css]]
{{colend}}

{{Multicol}}

==Subs==
* [[/Heaven energy/]]
* [[/Later/]]
* [[/More/]]
* [[/Ten/]]
* [[/Earning money/]]
* [[/Book/]]
* [[/Miscellaneous/]]
* [[/Areas/]]
* [[/Policy/]]
* [[/Navigating/]]
* [[/Added/]]
* [[/Learning/]]
* [[/Co-learners, co-researchers, co-teachers/]]
* [[/Templates/]]
* [[/Notes/]]
* [[/Drafts/]]
{{Multicol-break}}

==Miscellaneous==
* [[Special:Random]]
* [[Special:RandomRootpage/Topic]]
* [[Special:RandomRootpage/School]]
* [[Special:RandomRootpage/Portal]]
* [[Special:RandomRootpage/Draft]]
* [[Special:Random/Draft]]
* [[Special:RandomRootpage/Template]]
* [[Special:RandomRootpage]]
* [https://en.wikiversity.org/w/index.php?title=Special%3APrefixIndex&amp;prefix=&amp;namespace=104 All Topics]
* [https://en.wikiversity.org/w/index.php?title=Special%3APrefixIndex&amp;prefix=&amp;namespace=100 All Schools]
* [https://en.wikiversity.org/w/index.php?title=Special%3APrefixIndex&amp;prefix=&amp;namespace=102 All Portals]
* [[Special:Statistics]]
* [[Wikiversity:Statistics]]
* [[Wiktionary:Cosmogony]] 
* [[Draft:Index]]
* [https://en.wikiversity.org/wiki/Special:AllPages?from=&amp;to=&amp;namespace=118 Drafts (list)]
{{Multicol-break}}

==Yep==
*[[Controversy]]
* [[Special:Random/Topic]]
* [[Template:Multicol]]
* [http://listen.hatnote.com/ Listen to Wikipedia]
* [https://github.audio/ Listen to GitHub]
* [[Special:UncategorizedPages]]
* [[Draft:Archive/2024]]
* ||||
{{Multicol-break}}

==More==


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[[User:Michael Ten/common.css]]

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hmm

* [[w:Morihei_Ueshiba#Works|The Art of Peace]] by [[q:Morihei Ueshiba|Morihei Ueshiba]]

==== Study and research interests ====
*How might large language models affect learning and research. Will LLM's eventually seen like calculators are in math and sciences now? But for everything (all subjects/topics, including math, physics, ethics, biology, psychology, chemistry, engineering, art)?

==== Ideas to possibly explore ====
* constructive feedback
:: potential search to start &quot;constructive feedback site:.edu&quot; (without quotes)
* Can this Solar System potentially support a prosperous human population of over 500 billion humans who can all live lives of abundance and post-scarcity?
* The integration between large language models and robotics.
* Are there any open source large language models that are able to continually integrate new information as a form of training or fine tuning?</text>
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  <page>
    <title>User:Atcovi/to do</title>
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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
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      <text bytes="2966" sha1="10hrvygcp5x1q34777bry4lguhg5zvz" xml:space="preserve">==Atcovi/to do==

=== Current Projects (2026) ===

* [[User:Atcovi/Journey to Clinical PhD]] - figuring this out; current life goal.

* &lt;s&gt;[[WikiJournal Preprints/Mental health in Sri Lanka]] (and later in August: [[User:Atcovi/APA2026 Abstract]])&lt;/s&gt; {{Done}}
** &lt;s&gt;[[User:Atcovi/WikiJournal Preprints/Mental health in Sri Lanka/Future Outlook]].&lt;/s&gt;
====Suicidology/Psychopathology Works====

* Develop resources related to [[suicidology]] (3 stress response systems? effects of catecholamines on suicidal ideation? neurobiology of suicidal ideation? relation between autobiographical memory and suicide?), expand [[wikipedia:Suicidology#Theories_of_suicide|Suicidology#Theories_of_suicide]] either through [[WikiJournal of Science]] or WP editing.

* [[WikiJournal Preprints/Suicide amongst refugees in Sweden]] [https://scholar.google.com/scholar?hl=en&amp;as_sdt=0%2C47&amp;as_ylo=2020&amp;as_yhi=2025&amp;q=Suicide+in+Sweden+refugees&amp;btnG=]
* Get [[User:Atcovi/Spring2024]] &amp; [[User:Atcovi/Psychopathology]] into the mainspace. Develop [[Child psychology]] &amp; [[User:Atcovi/PSYC318W]] into a complete course. Merge [[Validity]] into [[User:Atcovi/PSYC318W|PSYC318W]].
=====Wikiversity-Related Works=====
* Promote [[Help:Project boxes]], something very useful and unique to Wikiversity. Focus on trying to not only create more project boxes, but to define resource types used in project boxes.
**Ex, what is a [[:Category:Workshops|workshop]]? What differentiates between an [[Help:Essay|essay]] and a [[Help:Paper|paper]]? What differentiates between a [[Template:Notes|notes resource]] (that may be ''derived'' from a homework assignment) and a [[Help:Assignment|homework assignment]] [small note: this page seems to be created by accident and may need a revamp]?
* [[Wikiversity:Original research and scholarly standards]] &amp; improvements/proposals for [[Wikiversity:Original research]] (ex, [[Template:Original research]] should be a mandatory addition to original research on WV + a notice letting readers know that the work is not established science). Develop other pages related to research ethics, including [[Wikiversity:Research]] &amp; [[Wikiversity:Research ethics]].
** [[Wikiversity:Review board]] - should this be Wikiversity 'crats that review original research proposals? 
* [[Wikiversity:Verifiability]] - start heavily scrutinizing pages that don't meet this criteria. 
* [[Wikiversity:Artificial intelligence]] - &quot;substantial&quot;? What defines &quot;substantial&quot;? 
* Expand [[Wikiversity:Differences between Wikiversity and Wikipedia]]. 
{{Archive box|
{{center top}}'''[[User:Atcovi/to do|To do list]]'''{{center bottom}}
----
{{center top}}'''Archives'''{{center bottom}}
*[[User:Atcovi/to do/Current Projects/2026]]
*[[User:Atcovi/to do/Current Projects/2023]]
*[[User:Atcovi/to do/Current Projects/January 4, 2022]]
*[[User:Atcovi/to do/Current Projects/September 2017 - January 2018]]
*[[User:Atcovi/to do/Current Projects/2015]]
----
}}
[[Category:Atcovi's Work]]</text>
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  <page>
    <title>Futurism</title>
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      <minor />
      <comment>Atcovi moved page [[User:Marshallsumter/Futurism]] to [[Futurism]] without leaving a redirect: not a &quot;typical Marshallsumter page&quot; as this was heavily worked on by another user, who has expressed a desire for this page to be returned to the mainspace [or at least out of userspace]</comment>
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      <text bytes="6627" sha1="7cz7nmi997iz75gte2ue1vso4z4h6jm" xml:space="preserve">[[Image:Feed Your Mind Futurism.png|thumb|right|200px|This is an artist's impression of futurism or feed your mind, brain, futurism, future studies, futurology. Credit: [[User:Ps2045|Singularity Utopia]].]]
'''Futurism''' in a general sense is concern with events and trends of the future or which anticipate the future. It was also an art movement begun in Italy in 1909 to celebrate and incorporate the energy and dynamism of modern technology. The art movement apparently ended in 1918. Perhaps it was interrupted by World War I.
{{RightTOC}}

==Theoretical futurism==
'''Def.''' a &quot;study and prediction of possible futures&quot;&lt;ref name=FuturismWikt&gt;{{ cite web
|author=[[wikt:User:Imran~enwiktionary|Imran~enwiktionary]]
|title=futurism, In: ''Wiktionary''
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=1 September 2015
|url=https://en.wiktionary.org/wiki/futurism
|accessdate=2015-09-17 }}&lt;/ref&gt; is called '''futurism'''.

==The future of futurism==

&quot;By combining the best of three different approaches to futurism--crisis futurism, evolutionary futurism, and spiritual futurism--we can realize vast human potentials and, perhaps, even attain the next stage of human evolution.&quot;&lt;ref name=Hubbard&gt;{{ cite journal
|author=Barbara Marx Hubbard
|title=The Future of Futurism: Creating a New Synthesis
|journal=Futurist
|month=April
|year=1983
|volume=17
|issue=2
|pages=52-8
|url=http://eric.ed.gov/?id=EJ278746
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Before hominins had large brains they first started walking upright. To evolve larger brains we need to be taller with appropriate support structure or we break our necks ever more often. To evolve the ability to fly we need to become smaller, have a lot of holes in our bones and fit our brains into a much smaller space. Or, we could invent aircraft and spacecraft and enjoy more recreational activity. There are always bills to pay.

&quot;The effort to think systematically about the future began little more than a half-century ago, and the results so far have not been impressive. Today's futurists hope that more sophisticated methods will allow them to provide a better picture of what tomorrow may bring.&quot;&lt;ref name=Rejeski&gt;{{ cite journal
|author=David Rejeski and Robert L. Olson
|title=Has Futurism Failed?
|journal=The Wilson Quarterly
|month=Winter
|year=2006
|volume=30
|issue=1
|pages=14-21
|url=http://www.jstor.org/stable/40261340
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Part of the problem with accurately predicting or trying to predict the future is that hedging and investment specialists are more likely to try to own the future and its inventors and inventions to maximize their gain, especially the more accurate the predictions become.

==Readings and learning media==
===Wikipedia===
{{colbegin|3}}
* [[Wikipedia: Technological singularity|Technological singularity]]
* [[Wikipedia: Futures studies|Futures studies]]
* [[Wikipedia: Systems thinking|Systems thinking]]
* [[Wikipedia: Technology roadmap|Technology roadmap]]
* [[Wikipedia: Future workshop|Future workshop]]
* [[Wikipedia: Technology forecasting|Technology forecasting]]
* [[Wikipedia: Trend analysis|Trend analysis]]
* [[Wikipedia: Futurist|Futurist]]
* [[Wikipedia: Decentralized autonomous organization|Decentralized autonomous organization]]
* [[Wikipedia: InterPlanetary File System|InterPlanetary File System]]
* [[Wikipedia: Accelerating change|Accelerating change]]
* [[Wikipedia: List of emerging technologies|List of emerging technologies]]
* [[Wikipedia: Outline of futures studies|Outline of futures studies]]
* [[Wikipedia: Singularity University|Singularity University]] - How can [[business]] and futurism intersect? 
* [[Wikipedia: Artificial uterus|Artificial uterus]] - a potential future technology that could eliminate physical suffering from child birth and reduce [[political science|political conflicts]]. 
* [[Wikipedia: NASA Institute for Advanced Concepts|NASA Institute for Advanced Concepts]]
* [[Wikipedia: Ledger (journal)|Ledger (journal)]] - Peer-reviewed academic journal covering research on all aspects of cryptocurrencies and blockchain technology, including mathematics, computer science, engineering, law, economics and philosophy.
* [[w:List of hypothetical technologies|List of hypothetical technologies]]
* [[w:Post-scarcity economy|Post-scarcity economy]]
* [[w:The End of Work|The End of Work]]
* [[w:Post-work society|Post-work society]]
{{colend}}

==Discussion questions==
* How can a better understanding of futurism and futurology help to better and benefit humanity and society? 
* How do futurism, [[business]], and [[economics]] intersect? What research could [[WikiJournal of Business and Economics|be disseminated]] in relation to such intersections?
* How can futurism be utilized to help mitigate [[environmental problems]]?
* How can futurism be utilized to help address [[social problems]] in a successful way?

==Areas of interest/sub niches==
{{colbegin|3}}
* [[Cryonics]]
* [[Life extension]]
* [[Singularity]]
* [[Exponential technologies]]
* [[Strategies for Engineered Negligible Senescence]]
* [[Basic income]]
* [[Overcoming pro-aging mindsets]]
* [[Anthropomorphic robotics]]
* [[Crowd funding]]
* [[Wikipedia:Cultured meat]]
* [[Vertical farming]]
* [[Renewable energy]]
* [[Artificial intelligence]]
* [[Exponential technologies]]
* [[Asteroid mining]]
* [[3D printing]]
* [[High-temperature superconductivity]]
* [[Colonizing Mars]]
* [[Off Earth colonization]]
* [[Nanotechnology]]
* [[Technological automation]]
* [[Unmanned aerial systems|Drones]]
* [[Decentralized autonomous organization|Decentralized autonomous organizations]]
* [[Synthetic biology]]
* [[Quantum computing]]
* [[Envisioning Our Future]]
* [[Robotics]]
* [[Rejuvenation]]
* [[Blockchain]] 
* [[Cryptocurrencies]] [[Wikipedia:Cryptocurrency]]
* [[Cryptoeconomics]]
* [[Post scarcity]]
* [[Fusion energy]]
* [[Supercomputing]]/[[Supercomputers]]
{{colend}}

==See also==
* [[Technological singularity]]
* [[Introduction to Futures Studies]]

==External links==
{{Sisterprojectsearch}}
{{article}}
* [https://hpluspedia.org H+Pedia] - large transhumanist and futurist wiki from Humanity+
* [http://futurology.reddit.com/ Futurology on Reddit]
* [https://www.fhi.ox.ac.uk/ Future of Humanity Institute at Oxford University]
* [https://su.org/ Singularity University]
* [http://cryptoubi.reddit.com Discuss universal basic income implemented through cryptocurrencies on Reddit] or start a conversation about it at [[Cryptocurrency basic income]]

==References==
{{reflist}}</text>
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    <revision>
      <id>2829799</id>
      <parentid>2829796</parentid>
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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
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      <minor />
      <comment>/* See also */ +[[Futurism/Quiz]]</comment>
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      <text bytes="6647" sha1="1reirwll8u8fdwk2m1oum0c9i5v89u2" xml:space="preserve">[[Image:Feed Your Mind Futurism.png|thumb|right|200px|This is an artist's impression of futurism or feed your mind, brain, futurism, future studies, futurology. Credit: [[User:Ps2045|Singularity Utopia]].]]
'''Futurism''' in a general sense is concern with events and trends of the future or which anticipate the future. It was also an art movement begun in Italy in 1909 to celebrate and incorporate the energy and dynamism of modern technology. The art movement apparently ended in 1918. Perhaps it was interrupted by World War I.
{{RightTOC}}

==Theoretical futurism==
'''Def.''' a &quot;study and prediction of possible futures&quot;&lt;ref name=FuturismWikt&gt;{{ cite web
|author=[[wikt:User:Imran~enwiktionary|Imran~enwiktionary]]
|title=futurism, In: ''Wiktionary''
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=1 September 2015
|url=https://en.wiktionary.org/wiki/futurism
|accessdate=2015-09-17 }}&lt;/ref&gt; is called '''futurism'''.

==The future of futurism==

&quot;By combining the best of three different approaches to futurism--crisis futurism, evolutionary futurism, and spiritual futurism--we can realize vast human potentials and, perhaps, even attain the next stage of human evolution.&quot;&lt;ref name=Hubbard&gt;{{ cite journal
|author=Barbara Marx Hubbard
|title=The Future of Futurism: Creating a New Synthesis
|journal=Futurist
|month=April
|year=1983
|volume=17
|issue=2
|pages=52-8
|url=http://eric.ed.gov/?id=EJ278746
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Before hominins had large brains they first started walking upright. To evolve larger brains we need to be taller with appropriate support structure or we break our necks ever more often. To evolve the ability to fly we need to become smaller, have a lot of holes in our bones and fit our brains into a much smaller space. Or, we could invent aircraft and spacecraft and enjoy more recreational activity. There are always bills to pay.

&quot;The effort to think systematically about the future began little more than a half-century ago, and the results so far have not been impressive. Today's futurists hope that more sophisticated methods will allow them to provide a better picture of what tomorrow may bring.&quot;&lt;ref name=Rejeski&gt;{{ cite journal
|author=David Rejeski and Robert L. Olson
|title=Has Futurism Failed?
|journal=The Wilson Quarterly
|month=Winter
|year=2006
|volume=30
|issue=1
|pages=14-21
|url=http://www.jstor.org/stable/40261340
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Part of the problem with accurately predicting or trying to predict the future is that hedging and investment specialists are more likely to try to own the future and its inventors and inventions to maximize their gain, especially the more accurate the predictions become.

==Readings and learning media==
===Wikipedia===
{{colbegin|3}}
* [[Wikipedia: Technological singularity|Technological singularity]]
* [[Wikipedia: Futures studies|Futures studies]]
* [[Wikipedia: Systems thinking|Systems thinking]]
* [[Wikipedia: Technology roadmap|Technology roadmap]]
* [[Wikipedia: Future workshop|Future workshop]]
* [[Wikipedia: Technology forecasting|Technology forecasting]]
* [[Wikipedia: Trend analysis|Trend analysis]]
* [[Wikipedia: Futurist|Futurist]]
* [[Wikipedia: Decentralized autonomous organization|Decentralized autonomous organization]]
* [[Wikipedia: InterPlanetary File System|InterPlanetary File System]]
* [[Wikipedia: Accelerating change|Accelerating change]]
* [[Wikipedia: List of emerging technologies|List of emerging technologies]]
* [[Wikipedia: Outline of futures studies|Outline of futures studies]]
* [[Wikipedia: Singularity University|Singularity University]] - How can [[business]] and futurism intersect? 
* [[Wikipedia: Artificial uterus|Artificial uterus]] - a potential future technology that could eliminate physical suffering from child birth and reduce [[political science|political conflicts]]. 
* [[Wikipedia: NASA Institute for Advanced Concepts|NASA Institute for Advanced Concepts]]
* [[Wikipedia: Ledger (journal)|Ledger (journal)]] - Peer-reviewed academic journal covering research on all aspects of cryptocurrencies and blockchain technology, including mathematics, computer science, engineering, law, economics and philosophy.
* [[w:List of hypothetical technologies|List of hypothetical technologies]]
* [[w:Post-scarcity economy|Post-scarcity economy]]
* [[w:The End of Work|The End of Work]]
* [[w:Post-work society|Post-work society]]
{{colend}}

==Discussion questions==
* How can a better understanding of futurism and futurology help to better and benefit humanity and society? 
* How do futurism, [[business]], and [[economics]] intersect? What research could [[WikiJournal of Business and Economics|be disseminated]] in relation to such intersections?
* How can futurism be utilized to help mitigate [[environmental problems]]?
* How can futurism be utilized to help address [[social problems]] in a successful way?

==Areas of interest/sub niches==
{{colbegin|3}}
* [[Cryonics]]
* [[Life extension]]
* [[Singularity]]
* [[Exponential technologies]]
* [[Strategies for Engineered Negligible Senescence]]
* [[Basic income]]
* [[Overcoming pro-aging mindsets]]
* [[Anthropomorphic robotics]]
* [[Crowd funding]]
* [[Wikipedia:Cultured meat]]
* [[Vertical farming]]
* [[Renewable energy]]
* [[Artificial intelligence]]
* [[Exponential technologies]]
* [[Asteroid mining]]
* [[3D printing]]
* [[High-temperature superconductivity]]
* [[Colonizing Mars]]
* [[Off Earth colonization]]
* [[Nanotechnology]]
* [[Technological automation]]
* [[Unmanned aerial systems|Drones]]
* [[Decentralized autonomous organization|Decentralized autonomous organizations]]
* [[Synthetic biology]]
* [[Quantum computing]]
* [[Envisioning Our Future]]
* [[Robotics]]
* [[Rejuvenation]]
* [[Blockchain]] 
* [[Cryptocurrencies]] [[Wikipedia:Cryptocurrency]]
* [[Cryptoeconomics]]
* [[Post scarcity]]
* [[Fusion energy]]
* [[Supercomputing]]/[[Supercomputers]]
{{colend}}

==See also==
* [[Futurism/Quiz]]
* [[Technological singularity]]
* [[Introduction to Futures Studies]]

==External links==
{{Sisterprojectsearch}}
{{article}}
* [https://hpluspedia.org H+Pedia] - large transhumanist and futurist wiki from Humanity+
* [http://futurology.reddit.com/ Futurology on Reddit]
* [https://www.fhi.ox.ac.uk/ Future of Humanity Institute at Oxford University]
* [https://su.org/ Singularity University]
* [http://cryptoubi.reddit.com Discuss universal basic income implemented through cryptocurrencies on Reddit] or start a conversation about it at [[Cryptocurrency basic income]]

==References==
{{reflist}}</text>
      <sha1>1reirwll8u8fdwk2m1oum0c9i5v89u2</sha1>
    </revision>
    <revision>
      <id>2829824</id>
      <parentid>2829799</parentid>
      <timestamp>2026-08-31T04:06:45Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
      </contributor>
      <comment>/* Discussion questions */ added</comment>
      <origin>2829824</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6896" sha1="nfrapeagfb6lokqln0tu7177plgm5cc" xml:space="preserve">[[Image:Feed Your Mind Futurism.png|thumb|right|200px|This is an artist's impression of futurism or feed your mind, brain, futurism, future studies, futurology. Credit: [[User:Ps2045|Singularity Utopia]].]]
'''Futurism''' in a general sense is concern with events and trends of the future or which anticipate the future. It was also an art movement begun in Italy in 1909 to celebrate and incorporate the energy and dynamism of modern technology. The art movement apparently ended in 1918. Perhaps it was interrupted by World War I.
{{RightTOC}}

==Theoretical futurism==
'''Def.''' a &quot;study and prediction of possible futures&quot;&lt;ref name=FuturismWikt&gt;{{ cite web
|author=[[wikt:User:Imran~enwiktionary|Imran~enwiktionary]]
|title=futurism, In: ''Wiktionary''
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=1 September 2015
|url=https://en.wiktionary.org/wiki/futurism
|accessdate=2015-09-17 }}&lt;/ref&gt; is called '''futurism'''.

==The future of futurism==

&quot;By combining the best of three different approaches to futurism--crisis futurism, evolutionary futurism, and spiritual futurism--we can realize vast human potentials and, perhaps, even attain the next stage of human evolution.&quot;&lt;ref name=Hubbard&gt;{{ cite journal
|author=Barbara Marx Hubbard
|title=The Future of Futurism: Creating a New Synthesis
|journal=Futurist
|month=April
|year=1983
|volume=17
|issue=2
|pages=52-8
|url=http://eric.ed.gov/?id=EJ278746
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Before hominins had large brains they first started walking upright. To evolve larger brains we need to be taller with appropriate support structure or we break our necks ever more often. To evolve the ability to fly we need to become smaller, have a lot of holes in our bones and fit our brains into a much smaller space. Or, we could invent aircraft and spacecraft and enjoy more recreational activity. There are always bills to pay.

&quot;The effort to think systematically about the future began little more than a half-century ago, and the results so far have not been impressive. Today's futurists hope that more sophisticated methods will allow them to provide a better picture of what tomorrow may bring.&quot;&lt;ref name=Rejeski&gt;{{ cite journal
|author=David Rejeski and Robert L. Olson
|title=Has Futurism Failed?
|journal=The Wilson Quarterly
|month=Winter
|year=2006
|volume=30
|issue=1
|pages=14-21
|url=http://www.jstor.org/stable/40261340
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Part of the problem with accurately predicting or trying to predict the future is that hedging and investment specialists are more likely to try to own the future and its inventors and inventions to maximize their gain, especially the more accurate the predictions become.

==Readings and learning media==
===Wikipedia===
{{colbegin|3}}
* [[Wikipedia: Technological singularity|Technological singularity]]
* [[Wikipedia: Futures studies|Futures studies]]
* [[Wikipedia: Systems thinking|Systems thinking]]
* [[Wikipedia: Technology roadmap|Technology roadmap]]
* [[Wikipedia: Future workshop|Future workshop]]
* [[Wikipedia: Technology forecasting|Technology forecasting]]
* [[Wikipedia: Trend analysis|Trend analysis]]
* [[Wikipedia: Futurist|Futurist]]
* [[Wikipedia: Decentralized autonomous organization|Decentralized autonomous organization]]
* [[Wikipedia: InterPlanetary File System|InterPlanetary File System]]
* [[Wikipedia: Accelerating change|Accelerating change]]
* [[Wikipedia: List of emerging technologies|List of emerging technologies]]
* [[Wikipedia: Outline of futures studies|Outline of futures studies]]
* [[Wikipedia: Singularity University|Singularity University]] - How can [[business]] and futurism intersect? 
* [[Wikipedia: Artificial uterus|Artificial uterus]] - a potential future technology that could eliminate physical suffering from child birth and reduce [[political science|political conflicts]]. 
* [[Wikipedia: NASA Institute for Advanced Concepts|NASA Institute for Advanced Concepts]]
* [[Wikipedia: Ledger (journal)|Ledger (journal)]] - Peer-reviewed academic journal covering research on all aspects of cryptocurrencies and blockchain technology, including mathematics, computer science, engineering, law, economics and philosophy.
* [[w:List of hypothetical technologies|List of hypothetical technologies]]
* [[w:Post-scarcity economy|Post-scarcity economy]]
* [[w:The End of Work|The End of Work]]
* [[w:Post-work society|Post-work society]]
{{colend}}

==Discussion questions==
* How can a better understanding of futurism and futurology help to better and benefit humanity and society? 
* How do futurism, [[business]], and [[economics]] intersect? What research could [[WikiJournal of Business and Economics|be disseminated]] in relation to such intersections?
* How can futurism be utilized to help mitigate [[environmental problems]]?
* How can futurism be utilized to help address [[social problems]] in a successful way?
* How is futurism conceptualized differently now than is was 5, 10, 15, 20 years ago? 
* What the potential implications of more individuals becoming aware of futurism as a method of analyzing trends and events as they relates to future happenings?

==Areas of interest/sub niches==
{{colbegin|3}}
* [[Cryonics]]
* [[Life extension]]
* [[Singularity]]
* [[Exponential technologies]]
* [[Strategies for Engineered Negligible Senescence]]
* [[Basic income]]
* [[Overcoming pro-aging mindsets]]
* [[Anthropomorphic robotics]]
* [[Crowd funding]]
* [[Wikipedia:Cultured meat]]
* [[Vertical farming]]
* [[Renewable energy]]
* [[Artificial intelligence]]
* [[Exponential technologies]]
* [[Asteroid mining]]
* [[3D printing]]
* [[High-temperature superconductivity]]
* [[Colonizing Mars]]
* [[Off Earth colonization]]
* [[Nanotechnology]]
* [[Technological automation]]
* [[Unmanned aerial systems|Drones]]
* [[Decentralized autonomous organization|Decentralized autonomous organizations]]
* [[Synthetic biology]]
* [[Quantum computing]]
* [[Envisioning Our Future]]
* [[Robotics]]
* [[Rejuvenation]]
* [[Blockchain]] 
* [[Cryptocurrencies]] [[Wikipedia:Cryptocurrency]]
* [[Cryptoeconomics]]
* [[Post scarcity]]
* [[Fusion energy]]
* [[Supercomputing]]/[[Supercomputers]]
{{colend}}

==See also==
* [[Futurism/Quiz]]
* [[Technological singularity]]
* [[Introduction to Futures Studies]]

==External links==
{{Sisterprojectsearch}}
{{article}}
* [https://hpluspedia.org H+Pedia] - large transhumanist and futurist wiki from Humanity+
* [http://futurology.reddit.com/ Futurology on Reddit]
* [https://www.fhi.ox.ac.uk/ Future of Humanity Institute at Oxford University]
* [https://su.org/ Singularity University]
* [http://cryptoubi.reddit.com Discuss universal basic income implemented through cryptocurrencies on Reddit] or start a conversation about it at [[Cryptocurrency basic income]]

==References==
{{reflist}}</text>
      <sha1>nfrapeagfb6lokqln0tu7177plgm5cc</sha1>
    </revision>
    <revision>
      <id>2829825</id>
      <parentid>2829824</parentid>
      <timestamp>2026-08-31T04:08:26Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
      </contributor>
      <comment>/* Discussion questions */ added</comment>
      <origin>2829825</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7052" sha1="k6khxbjdyw7x7nnecmt4nvwz9jtox6q" xml:space="preserve">[[Image:Feed Your Mind Futurism.png|thumb|right|200px|This is an artist's impression of futurism or feed your mind, brain, futurism, future studies, futurology. Credit: [[User:Ps2045|Singularity Utopia]].]]
'''Futurism''' in a general sense is concern with events and trends of the future or which anticipate the future. It was also an art movement begun in Italy in 1909 to celebrate and incorporate the energy and dynamism of modern technology. The art movement apparently ended in 1918. Perhaps it was interrupted by World War I.
{{RightTOC}}

==Theoretical futurism==
'''Def.''' a &quot;study and prediction of possible futures&quot;&lt;ref name=FuturismWikt&gt;{{ cite web
|author=[[wikt:User:Imran~enwiktionary|Imran~enwiktionary]]
|title=futurism, In: ''Wiktionary''
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=1 September 2015
|url=https://en.wiktionary.org/wiki/futurism
|accessdate=2015-09-17 }}&lt;/ref&gt; is called '''futurism'''.

==The future of futurism==

&quot;By combining the best of three different approaches to futurism--crisis futurism, evolutionary futurism, and spiritual futurism--we can realize vast human potentials and, perhaps, even attain the next stage of human evolution.&quot;&lt;ref name=Hubbard&gt;{{ cite journal
|author=Barbara Marx Hubbard
|title=The Future of Futurism: Creating a New Synthesis
|journal=Futurist
|month=April
|year=1983
|volume=17
|issue=2
|pages=52-8
|url=http://eric.ed.gov/?id=EJ278746
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Before hominins had large brains they first started walking upright. To evolve larger brains we need to be taller with appropriate support structure or we break our necks ever more often. To evolve the ability to fly we need to become smaller, have a lot of holes in our bones and fit our brains into a much smaller space. Or, we could invent aircraft and spacecraft and enjoy more recreational activity. There are always bills to pay.

&quot;The effort to think systematically about the future began little more than a half-century ago, and the results so far have not been impressive. Today's futurists hope that more sophisticated methods will allow them to provide a better picture of what tomorrow may bring.&quot;&lt;ref name=Rejeski&gt;{{ cite journal
|author=David Rejeski and Robert L. Olson
|title=Has Futurism Failed?
|journal=The Wilson Quarterly
|month=Winter
|year=2006
|volume=30
|issue=1
|pages=14-21
|url=http://www.jstor.org/stable/40261340
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Part of the problem with accurately predicting or trying to predict the future is that hedging and investment specialists are more likely to try to own the future and its inventors and inventions to maximize their gain, especially the more accurate the predictions become.

==Readings and learning media==
===Wikipedia===
{{colbegin|3}}
* [[Wikipedia: Technological singularity|Technological singularity]]
* [[Wikipedia: Futures studies|Futures studies]]
* [[Wikipedia: Systems thinking|Systems thinking]]
* [[Wikipedia: Technology roadmap|Technology roadmap]]
* [[Wikipedia: Future workshop|Future workshop]]
* [[Wikipedia: Technology forecasting|Technology forecasting]]
* [[Wikipedia: Trend analysis|Trend analysis]]
* [[Wikipedia: Futurist|Futurist]]
* [[Wikipedia: Decentralized autonomous organization|Decentralized autonomous organization]]
* [[Wikipedia: InterPlanetary File System|InterPlanetary File System]]
* [[Wikipedia: Accelerating change|Accelerating change]]
* [[Wikipedia: List of emerging technologies|List of emerging technologies]]
* [[Wikipedia: Outline of futures studies|Outline of futures studies]]
* [[Wikipedia: Singularity University|Singularity University]] - How can [[business]] and futurism intersect? 
* [[Wikipedia: Artificial uterus|Artificial uterus]] - a potential future technology that could eliminate physical suffering from child birth and reduce [[political science|political conflicts]]. 
* [[Wikipedia: NASA Institute for Advanced Concepts|NASA Institute for Advanced Concepts]]
* [[Wikipedia: Ledger (journal)|Ledger (journal)]] - Peer-reviewed academic journal covering research on all aspects of cryptocurrencies and blockchain technology, including mathematics, computer science, engineering, law, economics and philosophy.
* [[w:List of hypothetical technologies|List of hypothetical technologies]]
* [[w:Post-scarcity economy|Post-scarcity economy]]
* [[w:The End of Work|The End of Work]]
* [[w:Post-work society|Post-work society]]
{{colend}}

==Discussion questions==
* How can a better understanding of futurism and futurology help to better and benefit humanity and society? 
* How do futurism, [[business]], and [[economics]] intersect? What research could [[WikiJournal of Business and Economics|be disseminated]] in relation to such intersections?
* How can futurism be utilized to help mitigate [[environmental problems]]?
* How can futurism be utilized to help address [[social problems]] in a successful way?
* How is futurism conceptualized differently now than is was 5, 10, 15, 20 years ago? 
* What the potential implications of more individuals becoming aware of futurism as a method of analyzing trends and events as they relates to future happenings?
* What dynamics could occur so that advancing technology (automation, AI, robots) leads towards [[abundance]] instead of actual or artificial [[scarcity]]?

==Areas of interest/sub niches==
{{colbegin|3}}
* [[Cryonics]]
* [[Life extension]]
* [[Singularity]]
* [[Exponential technologies]]
* [[Strategies for Engineered Negligible Senescence]]
* [[Basic income]]
* [[Overcoming pro-aging mindsets]]
* [[Anthropomorphic robotics]]
* [[Crowd funding]]
* [[Wikipedia:Cultured meat]]
* [[Vertical farming]]
* [[Renewable energy]]
* [[Artificial intelligence]]
* [[Exponential technologies]]
* [[Asteroid mining]]
* [[3D printing]]
* [[High-temperature superconductivity]]
* [[Colonizing Mars]]
* [[Off Earth colonization]]
* [[Nanotechnology]]
* [[Technological automation]]
* [[Unmanned aerial systems|Drones]]
* [[Decentralized autonomous organization|Decentralized autonomous organizations]]
* [[Synthetic biology]]
* [[Quantum computing]]
* [[Envisioning Our Future]]
* [[Robotics]]
* [[Rejuvenation]]
* [[Blockchain]] 
* [[Cryptocurrencies]] [[Wikipedia:Cryptocurrency]]
* [[Cryptoeconomics]]
* [[Post scarcity]]
* [[Fusion energy]]
* [[Supercomputing]]/[[Supercomputers]]
{{colend}}

==See also==
* [[Futurism/Quiz]]
* [[Technological singularity]]
* [[Introduction to Futures Studies]]

==External links==
{{Sisterprojectsearch}}
{{article}}
* [https://hpluspedia.org H+Pedia] - large transhumanist and futurist wiki from Humanity+
* [http://futurology.reddit.com/ Futurology on Reddit]
* [https://www.fhi.ox.ac.uk/ Future of Humanity Institute at Oxford University]
* [https://su.org/ Singularity University]
* [http://cryptoubi.reddit.com Discuss universal basic income implemented through cryptocurrencies on Reddit] or start a conversation about it at [[Cryptocurrency basic income]]

==References==
{{reflist}}</text>
      <sha1>k6khxbjdyw7x7nnecmt4nvwz9jtox6q</sha1>
    </revision>
    <revision>
      <id>2829827</id>
      <parentid>2829825</parentid>
      <timestamp>2026-08-31T04:09:56Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
      </contributor>
      <comment>re-ordered.</comment>
      <origin>2829827</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7052" sha1="g0hydbpiewgm9u20rrbbx43skzks7my" xml:space="preserve">[[Image:Feed Your Mind Futurism.png|thumb|right|200px|This is an artist's impression of futurism or feed your mind, brain, futurism, future studies, futurology. Credit: [[User:Ps2045|Singularity Utopia]].]]
'''Futurism''' in a general sense is concern with events and trends of the future or which anticipate the future. It was also an art movement begun in Italy in 1909 to celebrate and incorporate the energy and dynamism of modern technology. The art movement apparently ended in 1918. Perhaps it was interrupted by World War I.
{{RightTOC}}

==Theoretical futurism==
'''Def.''' a &quot;study and prediction of possible futures&quot;&lt;ref name=FuturismWikt&gt;{{ cite web
|author=[[wikt:User:Imran~enwiktionary|Imran~enwiktionary]]
|title=futurism, In: ''Wiktionary''
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=1 September 2015
|url=https://en.wiktionary.org/wiki/futurism
|accessdate=2015-09-17 }}&lt;/ref&gt; is called '''futurism'''.

==The future of futurism==

&quot;By combining the best of three different approaches to futurism--crisis futurism, evolutionary futurism, and spiritual futurism--we can realize vast human potentials and, perhaps, even attain the next stage of human evolution.&quot;&lt;ref name=Hubbard&gt;{{ cite journal
|author=Barbara Marx Hubbard
|title=The Future of Futurism: Creating a New Synthesis
|journal=Futurist
|month=April
|year=1983
|volume=17
|issue=2
|pages=52-8
|url=http://eric.ed.gov/?id=EJ278746
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Before hominins had large brains they first started walking upright. To evolve larger brains we need to be taller with appropriate support structure or we break our necks ever more often. To evolve the ability to fly we need to become smaller, have a lot of holes in our bones and fit our brains into a much smaller space. Or, we could invent aircraft and spacecraft and enjoy more recreational activity. There are always bills to pay.

&quot;The effort to think systematically about the future began little more than a half-century ago, and the results so far have not been impressive. Today's futurists hope that more sophisticated methods will allow them to provide a better picture of what tomorrow may bring.&quot;&lt;ref name=Rejeski&gt;{{ cite journal
|author=David Rejeski and Robert L. Olson
|title=Has Futurism Failed?
|journal=The Wilson Quarterly
|month=Winter
|year=2006
|volume=30
|issue=1
|pages=14-21
|url=http://www.jstor.org/stable/40261340
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Part of the problem with accurately predicting or trying to predict the future is that hedging and investment specialists are more likely to try to own the future and its inventors and inventions to maximize their gain, especially the more accurate the predictions become.

==Discussion questions==
* How can a better understanding of futurism and futurology help to better and benefit humanity and society? 
* How do futurism, [[business]], and [[economics]] intersect? What research could [[WikiJournal of Business and Economics|be disseminated]] in relation to such intersections?
* How can futurism be utilized to help mitigate [[environmental problems]]?
* How can futurism be utilized to help address [[social problems]] in a successful way?
* How is futurism conceptualized differently now than is was 5, 10, 15, 20 years ago? 
* What the potential implications of more individuals becoming aware of futurism as a method of analyzing trends and events as they relates to future happenings?
* What dynamics could occur so that advancing technology (automation, AI, robots) leads towards [[abundance]] instead of actual or artificial [[scarcity]]?

==Areas of interest/sub niches==
{{colbegin|3}}
* [[Cryonics]]
* [[Life extension]]
* [[Singularity]]
* [[Exponential technologies]]
* [[Strategies for Engineered Negligible Senescence]]
* [[Basic income]]
* [[Overcoming pro-aging mindsets]]
* [[Anthropomorphic robotics]]
* [[Crowd funding]]
* [[Wikipedia:Cultured meat]]
* [[Vertical farming]]
* [[Renewable energy]]
* [[Artificial intelligence]]
* [[Exponential technologies]]
* [[Asteroid mining]]
* [[3D printing]]
* [[High-temperature superconductivity]]
* [[Colonizing Mars]]
* [[Off Earth colonization]]
* [[Nanotechnology]]
* [[Technological automation]]
* [[Unmanned aerial systems|Drones]]
* [[Decentralized autonomous organization|Decentralized autonomous organizations]]
* [[Synthetic biology]]
* [[Quantum computing]]
* [[Envisioning Our Future]]
* [[Robotics]]
* [[Rejuvenation]]
* [[Blockchain]] 
* [[Cryptocurrencies]] [[Wikipedia:Cryptocurrency]]
* [[Cryptoeconomics]]
* [[Post scarcity]]
* [[Fusion energy]]
* [[Supercomputing]]/[[Supercomputers]]
{{colend}}

==Readings and learning media==
===Wikipedia===
{{colbegin|3}}
* [[Wikipedia: Technological singularity|Technological singularity]]
* [[Wikipedia: Futures studies|Futures studies]]
* [[Wikipedia: Systems thinking|Systems thinking]]
* [[Wikipedia: Technology roadmap|Technology roadmap]]
* [[Wikipedia: Future workshop|Future workshop]]
* [[Wikipedia: Technology forecasting|Technology forecasting]]
* [[Wikipedia: Trend analysis|Trend analysis]]
* [[Wikipedia: Futurist|Futurist]]
* [[Wikipedia: Decentralized autonomous organization|Decentralized autonomous organization]]
* [[Wikipedia: InterPlanetary File System|InterPlanetary File System]]
* [[Wikipedia: Accelerating change|Accelerating change]]
* [[Wikipedia: List of emerging technologies|List of emerging technologies]]
* [[Wikipedia: Outline of futures studies|Outline of futures studies]]
* [[Wikipedia: Singularity University|Singularity University]] - How can [[business]] and futurism intersect? 
* [[Wikipedia: Artificial uterus|Artificial uterus]] - a potential future technology that could eliminate physical suffering from child birth and reduce [[political science|political conflicts]]. 
* [[Wikipedia: NASA Institute for Advanced Concepts|NASA Institute for Advanced Concepts]]
* [[Wikipedia: Ledger (journal)|Ledger (journal)]] - Peer-reviewed academic journal covering research on all aspects of cryptocurrencies and blockchain technology, including mathematics, computer science, engineering, law, economics and philosophy.
* [[w:List of hypothetical technologies|List of hypothetical technologies]]
* [[w:Post-scarcity economy|Post-scarcity economy]]
* [[w:The End of Work|The End of Work]]
* [[w:Post-work society|Post-work society]]
{{colend}}

==See also==
* [[Futurism/Quiz]]
* [[Technological singularity]]
* [[Introduction to Futures Studies]]

==External links==
{{Sisterprojectsearch}}
{{article}}
* [https://hpluspedia.org H+Pedia] - large transhumanist and futurist wiki from Humanity+
* [http://futurology.reddit.com/ Futurology on Reddit]
* [https://www.fhi.ox.ac.uk/ Future of Humanity Institute at Oxford University]
* [https://su.org/ Singularity University]
* [http://cryptoubi.reddit.com Discuss universal basic income implemented through cryptocurrencies on Reddit] or start a conversation about it at [[Cryptocurrency basic income]]

==References==
{{reflist}}</text>
      <sha1>g0hydbpiewgm9u20rrbbx43skzks7my</sha1>
    </revision>
    <revision>
      <id>2829828</id>
      <parentid>2829827</parentid>
      <timestamp>2026-08-31T04:10:57Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
      </contributor>
      <comment>put wikipedia links in related section.</comment>
      <origin>2829828</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7057" sha1="7l6ir9oq60osom4f79wynfdhdafosqr" xml:space="preserve">[[Image:Feed Your Mind Futurism.png|thumb|right|200px|This is an artist's impression of futurism or feed your mind, brain, futurism, future studies, futurology. Credit: [[User:Ps2045|Singularity Utopia]].]]
'''Futurism''' in a general sense is concern with events and trends of the future or which anticipate the future. It was also an art movement begun in Italy in 1909 to celebrate and incorporate the energy and dynamism of modern technology. The art movement apparently ended in 1918. Perhaps it was interrupted by World War I.
{{RightTOC}}

==Theoretical futurism==
'''Def.''' a &quot;study and prediction of possible futures&quot;&lt;ref name=FuturismWikt&gt;{{ cite web
|author=[[wikt:User:Imran~enwiktionary|Imran~enwiktionary]]
|title=futurism, In: ''Wiktionary''
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=1 September 2015
|url=https://en.wiktionary.org/wiki/futurism
|accessdate=2015-09-17 }}&lt;/ref&gt; is called '''futurism'''.

==The future of futurism==

&quot;By combining the best of three different approaches to futurism--crisis futurism, evolutionary futurism, and spiritual futurism--we can realize vast human potentials and, perhaps, even attain the next stage of human evolution.&quot;&lt;ref name=Hubbard&gt;{{ cite journal
|author=Barbara Marx Hubbard
|title=The Future of Futurism: Creating a New Synthesis
|journal=Futurist
|month=April
|year=1983
|volume=17
|issue=2
|pages=52-8
|url=http://eric.ed.gov/?id=EJ278746
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Before hominins had large brains they first started walking upright. To evolve larger brains we need to be taller with appropriate support structure or we break our necks ever more often. To evolve the ability to fly we need to become smaller, have a lot of holes in our bones and fit our brains into a much smaller space. Or, we could invent aircraft and spacecraft and enjoy more recreational activity. There are always bills to pay.

&quot;The effort to think systematically about the future began little more than a half-century ago, and the results so far have not been impressive. Today's futurists hope that more sophisticated methods will allow them to provide a better picture of what tomorrow may bring.&quot;&lt;ref name=Rejeski&gt;{{ cite journal
|author=David Rejeski and Robert L. Olson
|title=Has Futurism Failed?
|journal=The Wilson Quarterly
|month=Winter
|year=2006
|volume=30
|issue=1
|pages=14-21
|url=http://www.jstor.org/stable/40261340
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Part of the problem with accurately predicting or trying to predict the future is that hedging and investment specialists are more likely to try to own the future and its inventors and inventions to maximize their gain, especially the more accurate the predictions become.

==Discussion questions==
* How can a better understanding of futurism and futurology help to better and benefit humanity and society? 
* How do futurism, [[business]], and [[economics]] intersect? What research could [[WikiJournal of Business and Economics|be disseminated]] in relation to such intersections?
* How can futurism be utilized to help mitigate [[environmental problems]]?
* How can futurism be utilized to help address [[social problems]] in a successful way?
* How is futurism conceptualized differently now than is was 5, 10, 15, 20 years ago? 
* What the potential implications of more individuals becoming aware of futurism as a method of analyzing trends and events as they relates to future happenings?
* What dynamics could occur so that advancing technology (automation, AI, robots) leads towards [[abundance]] instead of actual or artificial [[scarcity]]?

==Areas of interest/sub niches==
{{colbegin|3}}
* [[Cryonics]]
* [[Life extension]]
* [[Singularity]]
* [[Exponential technologies]]
* [[Strategies for Engineered Negligible Senescence]]
* [[Basic income]]
* [[Overcoming pro-aging mindsets]]
* [[Anthropomorphic robotics]]
* [[Crowd funding]]
* [[Vertical farming]]
* [[Renewable energy]]
* [[Artificial intelligence]]
* [[Exponential technologies]]
* [[Asteroid mining]]
* [[3D printing]]
* [[High-temperature superconductivity]]
* [[Colonizing Mars]]
* [[Off Earth colonization]]
* [[Nanotechnology]]
* [[Technological automation]]
* [[Unmanned aerial systems|Drones]]
* [[Decentralized autonomous organization|Decentralized autonomous organizations]]
* [[Synthetic biology]]
* [[Quantum computing]]
* [[Envisioning Our Future]]
* [[Robotics]]
* [[Rejuvenation]]
* [[Blockchain]] 
* [[Cryptocurrencies]] 
* [[Cryptoeconomics]]
* [[Post scarcity]]
* [[Fusion energy]]
* [[Supercomputing]]/[[Supercomputers]]
{{colend}}

==Readings and learning media==
===Wikipedia===
{{colbegin|3}}
* [[Wikipedia: Technological singularity|Technological singularity]]
* [[Wikipedia: Futures studies|Futures studies]]
* [[Wikipedia: Systems thinking|Systems thinking]]
* [[Wikipedia: Technology roadmap|Technology roadmap]]
* [[Wikipedia: Future workshop|Future workshop]]
* [[Wikipedia: Technology forecasting|Technology forecasting]]
* [[Wikipedia: Trend analysis|Trend analysis]]
* [[Wikipedia: Cryptocurrency]]
* [[Wikipedia: Cultured meat]]
* [[Wikipedia: Futurist|Futurist]]
* [[Wikipedia: Decentralized autonomous organization|Decentralized autonomous organization]]
* [[Wikipedia: InterPlanetary File System|InterPlanetary File System]]
* [[Wikipedia: Accelerating change|Accelerating change]]
* [[Wikipedia: List of emerging technologies|List of emerging technologies]]
* [[Wikipedia: Outline of futures studies|Outline of futures studies]]
* [[Wikipedia: Singularity University|Singularity University]] - How can [[business]] and futurism intersect? 
* [[Wikipedia: Artificial uterus|Artificial uterus]] - a potential future technology that could eliminate physical suffering from child birth and reduce [[political science|political conflicts]]. 
* [[Wikipedia: NASA Institute for Advanced Concepts|NASA Institute for Advanced Concepts]]
* [[Wikipedia: Ledger (journal)|Ledger (journal)]] - Peer-reviewed academic journal covering research on all aspects of cryptocurrencies and blockchain technology, including mathematics, computer science, engineering, law, economics and philosophy.
* [[w:List of hypothetical technologies|List of hypothetical technologies]]
* [[w:Post-scarcity economy|Post-scarcity economy]]
* [[w:The End of Work|The End of Work]]
* [[w:Post-work society|Post-work society]]
{{colend}}

==See also==
* [[Futurism/Quiz]]
* [[Technological singularity]]
* [[Introduction to Futures Studies]]

==External links==
{{Sisterprojectsearch}}
{{article}}
* [https://hpluspedia.org H+Pedia] - large transhumanist and futurist wiki from Humanity+
* [http://futurology.reddit.com/ Futurology on Reddit]
* [https://www.fhi.ox.ac.uk/ Future of Humanity Institute at Oxford University]
* [https://su.org/ Singularity University]
* [http://cryptoubi.reddit.com Discuss universal basic income implemented through cryptocurrencies on Reddit] or start a conversation about it at [[Cryptocurrency basic income]]

==References==
{{reflist}}</text>
      <sha1>7l6ir9oq60osom4f79wynfdhdafosqr</sha1>
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  </page>
  <page>
    <title>Haskell programming in plain view</title>
    <ns>0</ns>
    <id>203942</id>
    <revision>
      <id>2829900</id>
      <parentid>2825741</parentid>
      <timestamp>2026-08-31T11:56:01Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2829900</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6273" sha1="duucy4wmynviu17q528rptiggp85dcq" xml:space="preserve">
==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]])
&lt;/br&gt;

==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]])

&lt;/br&gt;

==Using GHCi==
* Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]])
&lt;/br&gt;

==Using Libraries==
* Library ([[Media:Library.1.A.20170605.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;


==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]])
&lt;/br&gt;
&lt;/br&gt;

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.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.20260831.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

==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 &amp; 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 &amp; 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]])
&lt;/br&gt;

==Polymorphism==
* Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]])
&lt;/br&gt;

==Concurrent Haskell ==
&lt;/br&gt;


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]]</text>
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    <title>Futurism/Quiz</title>
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        <username>Atcovi</username>
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      <minor />
      <comment>Atcovi moved page [[User:Marshallsumter/Futurism/Quiz]] to [[Futurism/Quiz]] without leaving a redirect: not a &quot;typical Marshallsumter page&quot; as this was heavily worked on by another user, who has expressed a desire for this page to be returned to the mainspace [or at least out of userspace]</comment>
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      <text bytes="5547" sha1="f2enqnthwt4c63kbgo34ptp17iumfin" xml:space="preserve">[[Image:Feed Your Mind Futurism.png|thumb|right|250px|This is an artist's impression of futurism or feed your mind, brain, futurism, future studies, futurology. Credit: [[User:Ps2045|Singularity Utopia]].]]
'''[[Futurism]]''' is a lecture from the [[School:Economics|economics]] school about a specific economy, a predictable one.

You are free to take this quiz based on [[futurism]] at any time.

To improve your score, read and study the lecture, the links contained within, listed under [[Futurism/Quiz#See also|'''See also''']], [[Futurism/Quiz#External links|'''External links''']], in the {{tlx|income}} and {{tlx|economics resources}} templates. This should give you adequate background to get 100 %.

As a &quot;learning by doing&quot; resource, this quiz helps you to assess your knowledge and understanding of the information, and it is a quiz you may take over and over as a learning resource to improve your knowledge, understanding, test-taking skills, and your score.

'''Suggestion:''' Have the lecture available in a separate window.

To master the information and use only your memory while taking the quiz, try rewriting the information from more familiar points of view, or be creative with association.

Enjoy learning by doing!
{{clear}}

==Quiz==

&lt;quiz&gt;

{True or False, By combining the best of three different approaches to futurism--crisis futurism, evolutionary futurism, and spiritual futurism--we can realize vast human potentials and, perhaps, even attain the next stage of human evolution.
|type=&quot;()&quot;}
+ TRUE
- FALSE

{True or False, A dominant group regarding a futurism differs from a control group in that it rules the treatment of the control group.
|type=&quot;()&quot;}
+ TRUE
- FALSE

{Evidence that demonstrates that a model or idea for a futurism versus a control group is feasible is called a
|type=&quot;{}&quot;}
{ proof of concept (i) }.

{Yes or No, The effort to think systematically about the future began little more than a half-century ago, and the results so far have not been impressive.
|type=&quot;()&quot;}
+ Yes
- No

{True or False, A control group may be used in a futurism to demonstrate no effect or a standard effect versus a novel effort applied to a treatment group.
|type=&quot;()&quot;}
+ TRUE
- FALSE

{Complete the text:
|type=&quot;{}&quot;}
A short or { incomplete (i) } realization of a certain { method (i) } or idea to { demonstrate (i) } a treament's feasibility for any futurism is called a proof of { concept (i) }.

{Yes or No, Today's futurists hope that more sophisticated methods will allow them to provide a better picture of what tomorrow may bring.
|type=&quot;()&quot;}
+ Yes
- No

{True or False, Part of the problem with accurately predicting or trying to predict the future is that hedging and investment specialists are more likely to try to own the future and its inventors and inventions to maximize their gain, especially the more accurate the predictions become.
|type=&quot;()&quot;}
+ TRUE
- FALSE

{Complete the text:
|type=&quot;{}&quot;}
A proof-of-concept structure, including a control group, consists of { background (i) }, procedures, findings, and { interpretation (i) }.

{True or False, The purpose of a treatment group for a futurism is to describe natural processes or phenomena for the first time relative to a control group.
|type=&quot;()&quot;}
+ TRUE
- FALSE

&lt;/quiz&gt;

==Hypotheses==
{{main|Hypotheses}}
# A futurism economy can be a functioning economy when those endeavors which are profit-making are adequately balanced by the other endeavors which are not.

==See also==
{{div col|colwidth=20em}}
* [[Special:Search |Agriculture/Quiz]]
* [[Basic psychology Lecture 1 quiz]]
* [[Special:Search |Gift economy/Quiz]]
* [[Introduction to psychology/Psy102/Assessment/Quizzes]]
* [[Motivation and emotion/Assessment/Quizzes]]
* [[Special:Search |Pragmatics/Quiz]]
* [[Research purposes/Quiz]]
* [[Survey research and design in psychology/Assessment/Quizzes]]
{{Div col end}}

==External links==
* [http://www.ajol.info/ African Journals Online]
* [http://www.bing.com/search?q=&amp;go=&amp;qs=n&amp;sk=&amp;sc=8-15&amp;qb=1&amp;FORM=AXRE Bing Advanced search]
* [http://books.google.com/ Google Books]
* [http://scholar.google.com/advanced_scholar_search?hl=en&amp;lr= Google scholar Advanced Scholar Search]
* [http://www.jstor.org/ JSTOR]
* [http://www.lycos.com/ Lycos search]
* [http://www.ncbi.nlm.nih.gov/sites/gquery NCBI All Databases Search]
* [http://www.osti.gov/ Office of Scientific &amp; Technical Information]
* [http://psycnet.apa.org/ PsycNET]
* [http://www.questia.com/ Questia - The Online Library of Books and Journals]
* [http://online.sagepub.com/ SAGE journals online]
* [http://www.scirus.com/srsapp/advanced/index.jsp?q1= Scirus for scientific information only advanced search]
* [http://www.springerlink.com/ SpringerLink]
* [http://www.tandfonline.com/ Taylor &amp; Francis Online]
* [http://www.wikidoc.org/index.php/Main_Page WikiDoc The Living Textbook of Medicine]
* [http://onlinelibrary.wiley.com/advanced/search Wiley Online Library Advanced Search]
* [http://search.yahoo.com/web/advanced Yahoo Advanced Web Search]

&lt;!-- footer templates --&gt;
{{tlx|Anthropology resources}}{{Economics resources}}{{tlx|Humanities resources}}{{tlx|Income}}{{tlx|Terminology resources}}{{Sisterlinks|Gift economy}}

&lt;!-- categories --&gt;
[[Category:Anthropology quizzes]]
[[Category:Business quizzes]]
[[Category:Culture quizzes]]
[[Category:Economics quizzes]]
[[Category:Humanities quizzes]]
[[Category:Political science quizzes]]
[[Category:Resources last modified in January 2020]]
[[Category:Social psychology quizzes]]
[[Category:Sustainability quizzes]]</text>
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  <page>
    <title>User:Michael Ten/Later</title>
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      </contributor>
      <comment>update</comment>
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      <text bytes="2859" sha1="na29sjyz4ncswtt0otrhvfubzavkp3j" xml:space="preserve">Ideas to possibly develop or integrate into various areas later. 
==Some==
* [[Python]]
* [[Business/Earning money]]
* [[National University of Singapore]]
* [[Peking University]]
* [[Tsinghua University]]
* [[Archaeology of Specific Peoples and Civilizations]]
* [[Syllabi]]
* [[Teaching]]
* [[Analyzing research]]
* [[Research analysis]]
* [[Research studies]]
* [[Courses]]
* [[Learning]]
* [[Statistical analysis]]

* [[Peer to peer]]
* [[p2p]] / [[P2P]]
* [[Mesh networks]]
* [[Permissionless software]]
* [[CI/DC]]
* [[Coolify]]
* 

https://jme.bmj.com/content/44/11/751

==Some more - diversity of thought ==

* [[Diversity of thought]]
* [[w:Diversity ideologies|Diversity ideologies]]
* [[w:Diversity, equity, and inclusion|Diversity, equity, and inclusion]]
* [[w:Diversity (business)|Diversity (business)]]
* [[w:Cultural diversity|Cultural diversity]]
* [[w:Diversity, equity, and inclusion|Diversity, equity, and inclusion]]
* [[w:Neurodiversity|Neurodiversity]]
* [[w:Diversity (politics)|Diversity (politics)]]
* [[w:Unity in diversity|Unity in diversity]]
* [[w:Cultural diversity|Cultural diversity]]
* [[w:Gender diversity|Gender diversity]]
* [[w:Multicultural and diversity management|Multicultural and diversity management]]
* [[w:Team diversity|Team diversity]]
===external===
* [https://mtprof.msun.edu/Spr2007/fish.html Intellectual Diversity: What Is It and Do We Need It?]
* [https://roth.blogs.wesleyan.edu/tag/intellectual-diversity/ Intellectual Diversity]
* [https://tigerlearn.fhsu.edu/wp-content/uploads/Intellectual-Diversity-in-HE.pdf Intellectual Diversity in Higher Education]
* [https://yalebooks.yale.edu/2023/12/20/free-speech-and-intellectual-diversity/ Free Speech and Intellectual Diversity]
* [https://itsapps.odu.edu/ao/facultyhandbook/index.php?page=ch02s48.html Resolution Supporting Intellectual Diversity]
* [https://www.gsb.stanford.edu/exec-ed/difference/cognitive-diversity Cognitive Diversity: Know How to Harness It — And When to Rein It In — To Help Your Team Succeed]
* [https://online.merrimack.edu/cognitive-diversity-in-education/ Cognitive Diversity in Education]
* [https://sites.lsa.umich.edu/scottepage/research-2/diversity-research/ Diversity Research]
* [https://www.chapman.edu/diversity/news-and-events/cognitive-diversity/index.aspx Cognitive Diversity: Embracing Difference]
* [https://scholarworks.uttyler.edu/cgi/viewcontent.cgi?article=1005&amp;context=hrd_grad Enhancing the Cohesiveness of a Cognitive Diverse Team: the Role of Leadership]
* [https://docs.lib.purdue.edu/dissertations/AAI30505255/ Cognitive Diversity and Knowledge Integration in Student Design Teams]
** [https://www.proquest.com/docview/2838332291 Cognitive Diversity and Knowledge Integration in Student Design Teams]
* [https://cache.kzoo.edu/items/195fd798-9778-48f1-9fc5-d6c1dc47cec5 Modeling Cognitive Diversity]</text>
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    <title>Talk:Futurism</title>
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      <text bytes="1448" sha1="8ntq4u847oq640zcsasv0zx1fh1giqs" xml:space="preserve">==Space reduction==
:{{ping|User:Michael Ten}} We have tried a number of different space reductions at the beginning of resources. This version seems to work best! What do you think? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 23:15, 2 November 2016 (UTC)
::Looks good. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 05:08, 4 November 2016 (UTC)

== Draft ==

Regarding the move to Draft space, the resource was incorrectly categorized as a lecture. Lectures are not primary (main page) resources. I've cleaned up the categories and realigned the sections. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:29, 22 March 2018 (UTC)

== Why was this used to user space? ==
Why is this in userspace? Is there really no other name space? Main? Draft? etc where this can permanently stay and not be deleted? I think this page has valuable content, but i likely contributed so i am probably biased. Seeing pages like this moved to userspace or deleted is ''extremely demotivating'' in relation to created content for this wiki. limitless peace [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 18:55, 30 August 2026 (UTC)</text>
      <sha1>8ntq4u847oq640zcsasv0zx1fh1giqs</sha1>
    </revision>
    <revision>
      <id>2829795</id>
      <parentid>2829757</parentid>
      <timestamp>2026-08-31T00:18:39Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Why was this used to user space? */ Reply</comment>
      <origin>2829795</origin>
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      <text bytes="1855" sha1="4nr7a0yghr637yycdg3aghhwq88luvr" xml:space="preserve">==Space reduction==
:{{ping|User:Michael Ten}} We have tried a number of different space reductions at the beginning of resources. This version seems to work best! What do you think? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 23:15, 2 November 2016 (UTC)
::Looks good. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 05:08, 4 November 2016 (UTC)

== Draft ==

Regarding the move to Draft space, the resource was incorrectly categorized as a lecture. Lectures are not primary (main page) resources. I've cleaned up the categories and realigned the sections. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:29, 22 March 2018 (UTC)

== Why was this used to user space? ==
Why is this in userspace? Is there really no other name space? Main? Draft? etc where this can permanently stay and not be deleted? I think this page has valuable content, but i likely contributed so i am probably biased. Seeing pages like this moved to userspace or deleted is ''extremely demotivating'' in relation to created content for this wiki. limitless peace [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 18:55, 30 August 2026 (UTC)

:The page seemed to have been moved as part of an initiative to move Marshallsumter's pages out of the mainspace (which a lot of it was low-quality, but I don't think this page was properly reviewed), but per your desire I've moved it to the mainspace for you to work on. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:18, 31 August 2026 (UTC)</text>
      <sha1>4nr7a0yghr637yycdg3aghhwq88luvr</sha1>
    </revision>
    <revision>
      <id>2829797</id>
      <parentid>2829795</parentid>
      <timestamp>2026-08-31T00:19:26Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>Atcovi moved page [[User talk:Marshallsumter/Futurism]] to [[Talk:Futurism]] without leaving a redirect: not a &quot;typical Marshallsumter page&quot; as this was heavily worked on by another user, who has expressed a desire for this page to be returned to the mainspace [or at least out of userspace]</comment>
      <origin>2829795</origin>
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      <text bytes="1855" sha1="4nr7a0yghr637yycdg3aghhwq88luvr" xml:space="preserve">==Space reduction==
:{{ping|User:Michael Ten}} We have tried a number of different space reductions at the beginning of resources. This version seems to work best! What do you think? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 23:15, 2 November 2016 (UTC)
::Looks good. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 05:08, 4 November 2016 (UTC)

== Draft ==

Regarding the move to Draft space, the resource was incorrectly categorized as a lecture. Lectures are not primary (main page) resources. I've cleaned up the categories and realigned the sections. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:29, 22 March 2018 (UTC)

== Why was this used to user space? ==
Why is this in userspace? Is there really no other name space? Main? Draft? etc where this can permanently stay and not be deleted? I think this page has valuable content, but i likely contributed so i am probably biased. Seeing pages like this moved to userspace or deleted is ''extremely demotivating'' in relation to created content for this wiki. limitless peace [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 18:55, 30 August 2026 (UTC)

:The page seemed to have been moved as part of an initiative to move Marshallsumter's pages out of the mainspace (which a lot of it was low-quality, but I don't think this page was properly reviewed), but per your desire I've moved it to the mainspace for you to work on. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:18, 31 August 2026 (UTC)</text>
      <sha1>4nr7a0yghr637yycdg3aghhwq88luvr</sha1>
    </revision>
    <revision>
      <id>2829826</id>
      <parentid>2829797</parentid>
      <timestamp>2026-08-31T04:09:04Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
      </contributor>
      <comment>/* Why was this used to user space? */ ty ty</comment>
      <origin>2829826</origin>
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      <text bytes="2032" sha1="3rrw7cv1fkprg4q4do77tdt8vdk0wjl" xml:space="preserve">==Space reduction==
:{{ping|User:Michael Ten}} We have tried a number of different space reductions at the beginning of resources. This version seems to work best! What do you think? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 23:15, 2 November 2016 (UTC)
::Looks good. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 05:08, 4 November 2016 (UTC)

== Draft ==

Regarding the move to Draft space, the resource was incorrectly categorized as a lecture. Lectures are not primary (main page) resources. I've cleaned up the categories and realigned the sections. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:29, 22 March 2018 (UTC)

== Why was this used to user space? ==
Why is this in userspace? Is there really no other name space? Main? Draft? etc where this can permanently stay and not be deleted? I think this page has valuable content, but i likely contributed so i am probably biased. Seeing pages like this moved to userspace or deleted is ''extremely demotivating'' in relation to created content for this wiki. limitless peace [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 18:55, 30 August 2026 (UTC)

:The page seemed to have been moved as part of an initiative to move Marshallsumter's pages out of the mainspace (which a lot of it was low-quality, but I don't think this page was properly reviewed), but per your desire I've moved it to the mainspace for you to work on. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:18, 31 August 2026 (UTC)

:: Thanks ty. bless up 🐯 [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 04:09, 31 August 2026 (UTC)</text>
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  <page>
    <title>User talk:Marshallsumter/History</title>
    <ns>3</ns>
    <id>222410</id>
    <revision>
      <id>2829761</id>
      <parentid>2708265</parentid>
      <timestamp>2026-08-30T19:05:16Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
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      <text bytes="2755" sha1="knr0s5dqpjtr74ff0gsoolbg5wu5crx" xml:space="preserve">== Southern China IS NOT close to North Korea ==

As of 2017-04-22, the text includes a phrase, &quot;Southern China close to North Korea&quot;.  Southern China, as I understand that term, is over 2,000 km from North Korea.  At minimum, this phrase does not communicate well to me, but I don't know enough to fix it.  [[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 14:59, 22 April 2017 (UTC)

:I've added some files and text that may help! The phrase &quot;Southern China close to North Korea&quot; is in a quote from an author. To the people of the area north of Korea, Balhae was China, appears to be the author's intent. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 23:51, 22 April 2017 (UTC)

==Scope of ''History''==
It seems this draft is currently all over the place regarding the scope of history. My understanding is that &quot;History&quot; is primarily concerned with written records, so &quot;began&quot; with writing. That has been stretched at least in popular programs to include all of human activity (most of which is professionally considered &quot;pre-history&quot; and in the realm of archaeology or anthropology. Before ''Homo sapiens'', the subject becomes paleontology (and of course there's argument about that boundary). Rather than duplicate what's being done over at [[Draft:Paleontology]], might it be better to limit this resource to human history, subject to later refinement? {{Unsigned|Rhole2001|22 April 2018‎}}

:* [[Human history]] or [[History/Humans]] can be a separate resource. This lecture on [[History]] was put into &quot;Draft:&quot; namespace because it was in Main Page (Resource) space. See [[Wikiversity:Requests for Deletion#Main Page &quot;Lectures&quot;]] and [[Wikiversity talk:Requests for Deletion#Draft ns discussion]]. Lectures are different from encyclopedia articles. The beginning of this one has been contributed to by several. Again, this is kind of a TOC keynote lecture for the [[School:History]]. My major contribution is the timeline backwards from [[Recent history]] to the [[Hadean]] based on [[Draft:Geochronology]]. See also, [[Draft:Middle Ages]]. Have fun! --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 12:24, 23 April 2018 (UTC)

== Popularity ==

Firefox, Local

Wikiversity:Statistics/2020/01
:?.,750. ?,284


== why in user space ==

i am confused why this is not appropriate for somewhere in the main namespace... even if it has a page title change.  [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 19:05, 30 August 2026 (UTC)</text>
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  <page>
    <title>OpenStax</title>
    <ns>0</ns>
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      <id>2829852</id>
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      <contributor>
        <username>Andy?yes</username>
        <id>3006471</id>
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      <text bytes="3343" sha1="nlndvmzondfenousp21tru5tbvm88wv" xml:space="preserve">'''OpenStax''' (formerly OpenStax College) is a nonprofit ed-tech initiative based at Rice University. Since 2012, OpenStax has created peer-reviewed, openly licensed textbooks, which are available as free downloadable PDFs, web versions, audiobooks&lt;ref&gt;{{Cite web|url=https://openstax.org/blog/guest-post-how-audio-technology-is-creating-more-inclusive-learning|title=OpenStax {{!}} How audio technology is creating more inclusive learning|website=openstax.org|language=en-US|access-date=2025-10-20}}&lt;/ref&gt; and for a low cost in print. All textbook content is licensed under Creative Commons Attribution Licenses; specifically, the books are available under the Creative Commons Attribution-NonCommercial-ShareAlike License v4.0, which means that instructors are free to use, adapt, and remix the content, as long as they attribute OpenStax.&lt;ref&gt;[[Wikipedia: OpenStax]]&lt;/ref&gt;

The following Wikiversity resources devoted to OpenStax textbooks.  These resources also included materials available at [https://openstax.org/ '''openstax.org''']. Although some versions found on Wikiversity are out-of-date, some might find them more convenient to access.

*[[OpenStax American Government 3e]]
*[[OpenStax American Government 4e]]
*[[OpenStax University Physics|OpenStax University Physics (click to visit)]]

*[[OpenStax College Physics|OpenStax College Physics (click to visit)]]

*[[OpenStax Astronomy|OpenStax Astronomy (click to visit)]]
*[[OpenStax Astronomy 2e|OpenStax Astronomy 2e (click to visit)]]
*[[OpenStax Anatomy and Physiology 2e|OpenStax Anatomy &amp; Physiology 2e]]
*[[OpenStax Biology 2e]]
*[[OpenStax Business Ethics]]
*[[OpenStax Clinical Nursing Skills]]
*[[OpenStax College Success]]
*[[OpenStax College Success Concise]]
*[[OpenStax Concepts of Biology]]
*[[OpenStax Fundamentals of Nursing]]
*[[OpenStax Introduction to Anthropology]]
*[[OpenStax Introduction to Business]]
*[[OpenStax Introduction to Business 2e]]
*[[OpenStax Introduction to Political Science]]
*[[OpenStax Introduction to Sociology 3e]]
*[[OpenStax Lifespan Development]]
*[[OpenStax Maternal Newborn Nursing]]
*[[OpenStax Nutrition for Nurses]]
*[[OpenStax Organizational Behavior]]
*[[OpenStax Principles of Economics 3e]]
*[[OpenStax Principles of Macroeconomics 3e]]
*[[OpenStax Principles of Microeconomics 3e]]
*[[OpenStax Psychology 2e]]
*[[OpenStax US History]]
*[[OpenStax world history volume 1 to 1500|OpenStax World History, Volume 1: to 1500]]

*'''OpenStax Calculus: ''' No resources have been developed, but (out-of-date) pdf versions of the three volume textbook are posted on Wikiversity at: '''[[:File:CalculusVolume1-OP.pdf|V1]]''' | '''[[:File:CalculusVolume2-OP.pdf|V2]]''' | '''[[:File:CalculusVolume3-LR.pdf|V3]]''' 

== See Also ==
* [[Wikipedia: OpenStax]]

* [https://openstax.org/ OpenStax.org]
* [https://audileo.com/ Official OpenStax Audio Textbooks]

*[[:Category:openstax textbook]]
*[https://www.facebook.com/openstax/ OpenStax Facebook page]
* [https://www.ted.com/talks/richard_baraniuk_the_birth_of_the_open_source_learning_revolution TED Talk dated 2006-02] Founder Richard Baraniuk discussing Connexions
*[[Quizbank]]
* [https://www.youtube.com/watch?v=Xog2X2SnjvQ YouTube: Importing OpenStax content into Pressbooks]

== References ==
{{reflist}}

{{subpages/List}}

[[category:openstax file]] [[Category:Quizbank]]</text>
      <sha1>nlndvmzondfenousp21tru5tbvm88wv</sha1>
    </revision>
    <revision>
      <id>2829855</id>
      <parentid>2829852</parentid>
      <timestamp>2026-08-31T09:19:06Z</timestamp>
      <contributor>
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        <id>3006471</id>
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      <text bytes="3374" sha1="b6tae0oviw5kchdex6bvxhe2zcg470t" xml:space="preserve">'''OpenStax''' (formerly OpenStax College) is a nonprofit ed-tech initiative based at Rice University. Since 2012, OpenStax has created peer-reviewed, openly licensed textbooks, which are available as free downloadable PDFs, web versions, audiobooks&lt;ref&gt;{{Cite web|url=https://openstax.org/blog/guest-post-how-audio-technology-is-creating-more-inclusive-learning|title=OpenStax {{!}} How audio technology is creating more inclusive learning|website=openstax.org|language=en-US|access-date=2025-10-20}}&lt;/ref&gt; and for a low cost in print. All textbook content is licensed under Creative Commons Attribution Licenses; specifically, the books are available under the Creative Commons Attribution-NonCommercial-ShareAlike License v4.0, which means that instructors are free to use, adapt, and remix the content, as long as they attribute OpenStax.&lt;ref&gt;[[Wikipedia: OpenStax]]&lt;/ref&gt;

The following Wikiversity resources devoted to OpenStax textbooks.  These resources also included materials available at [https://openstax.org/ '''openstax.org''']. Although some versions found on Wikiversity are out-of-date, some might find them more convenient to access.

*[[OpenStax American Government 3e]]
*[[OpenStax American Government 4e]]
*[[OpenStax University Physics|OpenStax University Physics (click to visit)]]

*[[OpenStax College Physics|OpenStax College Physics (click to visit)]]

*[[OpenStax Astronomy|OpenStax Astronomy (click to visit)]]
*[[OpenStax Astronomy 2e|OpenStax Astronomy 2e (click to visit)]]
*[[OpenStax Anatomy and Physiology 2e|OpenStax Anatomy &amp; Physiology 2e]]
*[[OpenStax Biology 2e]]
*[[OpenStax Business Ethics]]
*[[OpenStax Clinical Nursing Skills]]
*[[OpenStax College Success]]
*[[OpenStax College Success Concise]]
*[[OpenStax Concepts of Biology]]
*[[OpenStax Entrepreneurship]]
*[[OpenStax Fundamentals of Nursing]]
*[[OpenStax Introduction to Anthropology]]
*[[OpenStax Introduction to Business]]
*[[OpenStax Introduction to Business 2e]]
*[[OpenStax Introduction to Political Science]]
*[[OpenStax Introduction to Sociology 3e]]
*[[OpenStax Lifespan Development]]
*[[OpenStax Maternal Newborn Nursing]]
*[[OpenStax Nutrition for Nurses]]
*[[OpenStax Organizational Behavior]]
*[[OpenStax Principles of Economics 3e]]
*[[OpenStax Principles of Macroeconomics 3e]]
*[[OpenStax Principles of Microeconomics 3e]]
*[[OpenStax Psychology 2e]]
*[[OpenStax US History]]
*[[OpenStax world history volume 1 to 1500|OpenStax World History, Volume 1: to 1500]]

*'''OpenStax Calculus: ''' No resources have been developed, but (out-of-date) pdf versions of the three volume textbook are posted on Wikiversity at: '''[[:File:CalculusVolume1-OP.pdf|V1]]''' | '''[[:File:CalculusVolume2-OP.pdf|V2]]''' | '''[[:File:CalculusVolume3-LR.pdf|V3]]''' 

== See Also ==
* [[Wikipedia: OpenStax]]

* [https://openstax.org/ OpenStax.org]
* [https://audileo.com/ Official OpenStax Audio Textbooks]

*[[:Category:openstax textbook]]
*[https://www.facebook.com/openstax/ OpenStax Facebook page]
* [https://www.ted.com/talks/richard_baraniuk_the_birth_of_the_open_source_learning_revolution TED Talk dated 2006-02] Founder Richard Baraniuk discussing Connexions
*[[Quizbank]]
* [https://www.youtube.com/watch?v=Xog2X2SnjvQ YouTube: Importing OpenStax content into Pressbooks]

== References ==
{{reflist}}

{{subpages/List}}

[[category:openstax file]] [[Category:Quizbank]]</text>
      <sha1>b6tae0oviw5kchdex6bvxhe2zcg470t</sha1>
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  <page>
    <title>User:JimKillock</title>
    <ns>2</ns>
    <id>247006</id>
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      <text bytes="689" sha1="etcwip6tjzvueka5ilmekyn277a1yzb" xml:space="preserve">Main work here: 
* [[Latin]] (not my course, just my import); thanks go to [[user:CarpeLanam]] for writing it
* [[Mundus Latinus: A Course in Practical Latin]]
* [[User:JimKillock/Nos in Schola Latine loquimur/Capitulum Secundum On being late]]


* [[User:JimKillock/Personal learning]]
* [[Colloquia familiara: a selection]]

* [[User:JimKillock/Latin AI prompts/Characters]]
* [[User:JimKillock/Latin drafts]]
See also:

* [[Portal:Latin]]

Other user pages:
* [[Wikisource:User:JimKillock]]  
* [[Wikisource:la:Usor:JimKillock]]  
* [[Wikibooks:User:JimKillock]]
* [[Wikipedia:User:JimKillock]]
* [[Wikidata:User:JimKillock]]
* [[Commons:User:JimKillock]]
* [[Wikidata:User:JimKillock]]</text>
      <sha1>etcwip6tjzvueka5ilmekyn277a1yzb</sha1>
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    <revision>
      <id>2829873</id>
      <parentid>2829846</parentid>
      <timestamp>2026-08-31T10:44:48Z</timestamp>
      <contributor>
        <username>JimKillock</username>
        <id>330368</id>
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      <text bytes="656" sha1="6f82z15erhdi9i6v255ur5vxv3kncrc" xml:space="preserve">Main work here: 
* [[Latin]] (not my course, just my import); thanks go to [[user:CarpeLanam]] for writing it
* [[Mundus Latinus: A Course in Practical Latin]]
* [[User:JimKillock/Nos in Schola Latine loquimur]]


* [[User:JimKillock/Personal learning]]
* [[Colloquia familiara: a selection]]

* [[User:JimKillock/Latin AI prompts/Characters]]
* [[User:JimKillock/Latin drafts]]
See also:

* [[Portal:Latin]]

Other user pages:
* [[Wikisource:User:JimKillock]]  
* [[Wikisource:la:Usor:JimKillock]]  
* [[Wikibooks:User:JimKillock]]
* [[Wikipedia:User:JimKillock]]
* [[Wikidata:User:JimKillock]]
* [[Commons:User:JimKillock]]
* [[Wikidata:User:JimKillock]]</text>
      <sha1>6f82z15erhdi9i6v255ur5vxv3kncrc</sha1>
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  </page>
  <page>
    <title>Social Victorians/People/Gwladys Robinson</title>
    <ns>0</ns>
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      <contributor>
        <username>Scogdill</username>
        <id>1331941</id>
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      <text bytes="105658" sha1="dqp2ynkkm6kvnvcbhldk7b2n8wyd288" xml:space="preserve">{{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==
== 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, &quot;The high top hat, usually black or dark gray, had reached its characteristic shape by 1798 and dominated the entire nineteenth century.&quot; (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, &lt;blockquote&gt;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 &quot;every&quot; man throughout the history of the world &quot;must, necessarily and habitually, wear some kind of covering to his head&quot;. 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: &quot;the nap rusty, the band a mournful strip of tarnished lace; but still a Hat&quot;, which &quot;stamps him as being associated, in however slender a manner, with a learned profession&quot;. 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.&lt;ref name=&quot;:23&quot;&gt;{{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}}&lt;/ref&gt; (509–511 [of 972]) &lt;/blockquote&gt;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.&lt;ref name=&quot;:4&quot;&gt;{{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}}&lt;/ref&gt; 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.&lt;ref name=&quot;g4223&quot;&gt;{{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=}}&lt;/ref&gt;
[[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.&lt;ref&gt;{{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}}&lt;/ref&gt;

== 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.&lt;ref name=&quot;:4&quot; /&gt;

== Cosmetics ==
[[Victorian-era cosmetics]] were typically minimal, as makeup was associated by the middle classes with promiscuity. However, small amounts of pale face powder or powdered blush were more widely used.&lt;ref&gt;{{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}}&lt;/ref&gt; Some cosmetics contained toxic or caustic ingredients like lead, mercury, ammonia, and arsenic {{Citation needed|date=October 2025}}.

Hair color

== Men's fashion ==
[[File:Mens Coats 1872 Fashion Plate.jpg|thumb|upright|Drawing of Victorian men 1870s]]

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. Waistcoats were single- or double-breasted, with shawl or notched collars, and might be finished in double points at the lowered waist. 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. Trousers had fly fronts, and [[breeches]] were used for formal functions and when horseback riding. Men wore [[top hat]]s, with wide brims in sunny weather.

=== 1850s ===
According to Judith Flanders,&lt;blockquote&gt;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.&lt;ref name=&quot;:23&quot; /&gt; (167–168 [of 972])&lt;/blockquote&gt;

==== Original Text ====
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 &quot;wings&quot;. The upper-class continued to wear top hats, and [[bowler hat]]s were worn by the working class.

=== 1860s ===
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. 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. Top hats briefly became the very tall &quot;stovepipe&quot; shape, but a variety of other hat shapes were popular.

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.

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.

Starting from the [[1890s in fashion|1890s]], the [[blazer]] was introduced, and was worn for sports, sailing, and other casual activities.&lt;ref&gt;{{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}}&lt;/ref&gt;

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.&lt;ref&gt;{{cite web|url=http://www.victorianweb.org/art/costume/nunn21.html|title=Victorian Men's Fashions, 1850–1900: Hair}}&lt;/ref&gt;

Distinguishing what men really wore from what was marketed to them in periodicals and advertisements is difficult, as reliable records do not exist.&lt;ref name=&quot;shannon597&quot;&gt;{{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}}&lt;/ref&gt;

=== Notes ===

* Shirts and collars separated, &quot;by 1827 separate collars became available&quot;  (Payne 460)
* transition from frock coats to ditto suits, 1850s (Payne, 463)
* 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)
* To correct and prevent errors being made in men's court dress, the Lord Chamberlain published &quot;a summary of regulations for court uniform and dress&quot; (in ''Dress Worn by Gentlemen at Her Majesty's Court'', 1875).&lt;ref&gt;{{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}}&lt;/ref&gt;
* Brent Shannon. &quot;Refashioning Men: Fashion, Masculinity, and the Cultivation of the Male Consumer in Britain, 1860–1914.&quot; Victorian Studies 46, no. 4 (Summer 2004): 597–630.
* In the context of 19th-c evolving definitions of gender, both femininity and masculinity evolved as concepts. The changing definitions of masculinity affected men's clothing.


Carolyn Kirby:&lt;blockquote&gt;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.&lt;ref&gt;{{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}}&lt;/ref&gt;&lt;/blockquote&gt;David Kuchta:&lt;blockquote&gt;... 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.&lt;ref&gt;{{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}}&lt;/ref&gt; (2)&lt;/blockquote&gt;Brent Shannon:&lt;blockquote&gt;&quot;Costume,&quot; wrote Max Beerbohm in 1896, &quot;enables us to classify any 'professional man' at a glance, be he lawyer, leech or who not&quot; (24–25). A man's profession and class were read by his jacket, his hat, what he rode in, and how he carried himself. &quot;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,&quot; conduct author John Wanamaker confessed; &quot;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&quot; (1).&lt;sup&gt;11&lt;/sup&gt;

Such assertions were predicated on the powerful Victorian conviction that outward appearance reflected inner qualities.&lt;ref&gt;{{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}}&lt;/ref&gt; (148)&lt;/blockquote&gt;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):&lt;blockquote&gt;The Industrial Revolution that began in the late 18th century had a great impact on the development of fashion in the 19th century, there was a clear change in men's clothing at the beginning of the 19th century, not only in style but also in the appearance of the English sewing machine, and from this date, English clothing became world-class, and this was not only for England but for all of Europe is undoubtedly due to the French Revolution that stripped Europe of its previous leadership of fashion, so the 19th century belonged to the English in terms of fashion [16].

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. Neoclas- sicism 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].

Men's clothing during this century consisted of black, brown, blue (dark, shiny, or bright), olive green, and grey. The preferred beautiful colors for evening wear were blue, followed by brown and green, while the fabrics for summer trousers were dark grey or black (with blue coats), and for daywear were light colors such as white or beige (Hussein, T. 2002). [16].

A study: (Historical, and Cultural Impact on the Costume Development) showed that depending on the functional and aesthetic characteristics, the division of clothing according to gender and age continued for centuries, whether informal or ceremonial, and varied according to gender, general style, nature of the jewelry, as well as family status, and stated that the traditional costume indirectly linked man to nature, as it was a gateway to the relationship between the body (the small world) and the world (the big world) [20].&lt;ref&gt;{{Cite journal|last=Alzahrani|first=Sarah Gharmallah|last2=Saroukh|first2=Safia Abdelaziz|date=2024|title=The Semiotic Dimension of Men's Fashion in Modern Eras|url=http://www.sciencepg.com/journal/ijla|journal=International Journal of Literature and Arts|volume=Vol. 12, No. 5|via=Research Gate}}&lt;/ref&gt; (136)

# [16]  Hussein, T. (2002). The History and Development of Fashion „Part III‟ Modern Times, Nahdet Misr for Printing and Publishing, Cairo.
# [17]  Franklin, H. (Aug 18, 2020). Published on Jun 25, 2020, Retrieved: &lt;nowiki&gt;https://fashionhistory.fitnyc.edu/1800-1809/&lt;/nowiki&gt; 11/11/2023. Edited. ...
# [20] Park, S.J., &amp; Park, K.S. (2006). Semiotic Analysis on Advertisement Expression of Men's Toiletries. The Research Journal of the Costume Culture, 14(2), 234-246.
&lt;/blockquote&gt;

=== Albert Edward, Prince of Wales ===
Influence of Bertie, Albert Edward, Prince of Wales

McNeil:&lt;blockquote&gt;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. 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.” 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 bot- toms at a dirty racing track) and, as his girth grew, undoing the bottom button of his waistcoat.&lt;ref&gt;McNeil, Peter. &quot;Men's Fashion: 1800–2022.&quot; 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.&lt;/ref&gt;&lt;/blockquote&gt;Albert Edward, Prince of Wales, very concerned with fashion and authoritative about it.

Virginia Cowles:&lt;blockquote&gt;It would be wrong to give the impression that the Heir Apparent was unhappy. If he could not work, at least he could play, and he did this very well. He loved being royal. He revelled in the rank and authority and privilege and luxury that accompanied the role of Prince of Wales. There were radicals who liked to lampoon him , and courtiers who wanted to reform him. But there was a much bigger group, a rich, fashionable, powerful society who adored him, fawned on him, gratified him, and copied everything he did.

Paradoxically this adulation often increased the Prince’s freedom of movement. A contemporary writer states that it was possible for the Prince of Wales to walk along Piccadilly, or St. James’ Street or Pall Mall without being recognized. Why? Because photography was still undeveloped? Oh no. It was due to ‘the curious fact that there are in society several gentlemen who bear an extraordinary resemblance to him, and who take some pride in dressing and moving exactly like him, so that it is often very difficult to identify him as he passes in the street on foot or in a hansom cab.

But the vogue of imitating the Prince did not stop at his beard, his clothes and his walk. Once when he had an attack of rheumatism in his shoulder, he was obliged to shake hands with his expo pressed stiffly to his side. Immediately this peculiar hand-shake was adopted by fashionable London. And when Alexandra [128–129] had a severe illness in the late sixties which left her lame for life, the smartest ladies in the land began to walk with a slightly halting gait, which became known as ‘the Alexandra Limp’.

The aping of royalty was not considered vulgar. On the whole the Prince and Princess were amused and flattered by it, but every now and then someone went too far. On one occasion a rich manufacturer from the North drove in the Park with his horses wearing headbands of the royal scarlet used exclusively by the Prince. The Heir Apparent did not attempt to hide his displeasure. His blue eyes grew cold, and his lower lip protruded in the famous Guelph pout. As a sharp lesson to the perpetrators of this unforgivably bad taste he drove in the ZPark the next day with his horses wearing black headbands. The manufacturer’s wife and daughters could not fail to observe the significance of this slight, and left the Park in tears; and the Prince’s friends congratulated him on his clever rebuff.

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.

As the years rolled on the Prince became an ever-increasing authority on dress. Tailors from all over Europe used to gather to study his clothes. Their favorite meeting place was Homburg, and later, Marienbad. Here they could catch a glimpse of the Prince half a dozen times a day, strolling along the promenade, or riding in an open carriage. Once Bertie dressed hurriedly and forgot to fasten the last button on his waistcoat; this became a permanent fashion.

British manufacturers were not slow to realise what an asset they had in the Heir Apparent and kept a vigilant eye on his movements. Once, one of them declared in outraged tones that he was buying his gloves in France. A storm blew up of such proportions that the Prince’s secretary, Sir Francis Knollys, was forced to make a statement to the press. First, he declared that the Prince always had his gloves made in England, and second (and this was calculated to silence the critics) that His Royal Highness was very economical in the use of gloves and only found it necessary to order two dozen pairs a year.

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  &quot;heavy swells&quot; 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 &quot;Glossy Top&quot;. He is responsible for the top hat as we know it today.&lt;ref&gt;{{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/}}&lt;/ref&gt; (128–131)&lt;/blockquote&gt;

==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]].&lt;ref&gt;{{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}}&lt;/ref&gt;

The mourning dress worn by Queen Victoria (below, right) &quot;shows the traditional touches of mourning attire, which she wore from the death of her husband, Prince Albert (1819–1861), until her own death.&quot;&lt;ref&gt;{{Cite web|url=https://www.metmuseum.org/art/collection/search/155839?&amp;searchField=All&amp;sortBy=Relevance&amp;deptids=8&amp;ft=queen+victoria&amp;offset=0&amp;rpp=20&amp;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}}&lt;/ref&gt; 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 &amp; Co., 1887) gives clear instructions, such as the following:&lt;ref&gt;{{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}}&lt;/ref&gt;

{| class=&quot;wikitable&quot;
|-
! 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 &quot;connections&quot; (who were entitled to ordinary mourning for a period of &quot;1–3 weeks, depending on level of intimacy&quot;). 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.&lt;ref&gt;{{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}}&lt;/ref&gt; 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.&lt;ref&gt;{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|page=341}}&lt;/ref&gt;

== 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, &quot;by the early 1850s there were thousands of steam-powered looms churning out millions of miles of fabric every year&quot;  [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) &quot;wash[ed] the fashionable landscape in a haze of purple.&quot;&lt;ref name=&quot;:22&quot;&gt;{{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}}&lt;/ref&gt; (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 &quot;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.&quot;&lt;ref name=&quot;:22&quot; /&gt; (255) Perhaps the most famous of these is arsenic green, used on fabrics, wallpapers, and trim: &quot;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.&quot;&lt;ref name=&quot;:22&quot; /&gt; (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 &quot;lightweight domestic machines&quot; for home sewing, radically increasing women's control over their own dress.&lt;ref name=&quot;:24&quot;&gt;{{Cite book|title=Victorian Fashions for Women|last=Kay|first=Fiona|last2=Storey|first2=Neil R.|publisher=Pen &amp; Sword History|year=2022|isbn=978 1 39900 416 9|location=Yorkshire and Philadelphia|pages=}}&lt;/ref&gt; (91 [of 298])
* the spread of journalism for women and fashion journalism

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.&lt;ref name=&quot;:3&quot;&gt;{{Cite book|title=The Culture of Fashion|last=Breward|first=Christopher|publisher=Manchester University Press|year=1995|pages=145–180}}&lt;/ref&gt; 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.&lt;ref&gt;{{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}}&lt;/ref&gt; 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.&lt;ref&gt;{{cite book|last1=Shrimpton|first1=J|title=Victorian Fashion|publisher=Bloomsbury Shire Publications}}&lt;/ref&gt; 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.&lt;ref&gt;{{cite book|last1=Aspelund|first1=Karl|title=Fashioning Society|publisher=Fairchild Books}}&lt;/ref&gt;

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.&lt;ref name=&quot;Haute Couture&quot;&gt;{{cite book|last1=Martin|first1=Richard|last2=Koda|first2=Harold|title=Haute Couture|publisher=The Metropolitan Museum of Art}}&lt;/ref&gt; 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.&lt;ref&gt;{{cite book|last1=Saillard|first1=Olivier|last2=Zazzo|first2=Anne|title=Paris Haute Couture|publisher=Skira Flammarion}}&lt;/ref&gt;

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.&lt;ref name=&quot;Haute Couture&quot;/&gt;

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|&quot;The proper length for little girls' skirts at various ages&quot;, 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.&lt;ref name=&quot;:21&quot;&gt;{{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=}}&lt;/ref&gt; (71 [of 1182]) No evidence exists that tight lacing was widespread or particularly dangerous.&lt;ref name=&quot;:21&quot; /&gt; ()


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.&lt;ref name=&quot;shannon597&quot;/&gt; 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.&lt;ref&gt;{{cite book |last=Gernsheim |first=Alison |title=Victorian &amp; Edwardian Fashion: A Photographic Survey |year=1981 |publisher=Dover Publications |location=New York |page=65|edition=New |isbn=0-486-24205-6}}&lt;/ref&gt;

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.&lt;ref&gt;{{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&amp;pg=PA246}} From pp. 246-247:  &quot;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!&quot;&lt;/ref&gt;

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 &quot;A Diary in America&quot;]
* [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 &amp; 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 &quot;unmentionables&quot;). The discussion of such a topic, it was feared, would gravitate towards unhealthy attention on anatomical details. As one Victorian lady expressed it: &quot;[those] are not things, my dear, that we speak of; indeed, we try not even to think of them&quot;, in contrast to current norms.&lt;ref&gt;{{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}}&lt;/ref&gt; 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&quot;.&lt;ref&gt;{{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}}&lt;/ref&gt;}}

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.&lt;ref&gt;{{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}}&lt;/ref&gt;

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.&lt;ref name=&quot;O'Connor&quot;/&gt;

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 &quot;her feet are improperly protected, and she wears no flannel petticoat or woollen stockings&quot;.&lt;ref name=&quot;O'Connor&quot;/&gt;
[[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 &quot;[[divided skirt]]&quot;, 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 &quot;unwomanly&quot; association of the style with the American [[Bloomers]] movement.&lt;ref&gt;{{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}}&lt;/ref&gt; [[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 &quot;full circle&quot; 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]],&lt;ref&gt;{{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}}&lt;/ref&gt; &quot;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,&quot; said Koda, &quot;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.&quot;

====Evolution of Victorian women's fashion====
&lt;gallery&gt;
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.
&lt;/gallery&gt;


{{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 (&lt;small&gt;Irish:&lt;/small&gt; [/ˈda.rax/ /oː/ /sˠkˠoː/l̪ˠəi/]; born 1963&lt;ref&gt;{{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}}&lt;/ref&gt;) 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”&lt;ref&gt;{{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}}&lt;/ref&gt; and &quot;one of the great Irish language novelists [duine d’úrscéalaithe móra na Gaeilge].&quot;&lt;ref name=&quot;:17&quot; /&gt; His writing has been called “the high literature of the Irish language.”&lt;ref&gt;Ó Coimín, Maitiú. ''Nós'' 2 February 2018). Qtd. in &quot;Táin Bó Cuailnge.&quot; ''Leabhar Breac''. Retrieved 25 August 2025.&lt;/ref&gt;

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].”&lt;ref name=&quot;:11&quot;&gt;{{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}}&lt;/ref&gt; 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).&lt;ref name=&quot;:7&quot;&gt;{{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}}&lt;/ref&gt;&lt;ref&gt;{{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}}&lt;/ref&gt; 

He graduated the [[wikipedia:University_of_Galway|University of Galway]] (then University College Galway) with a B.A. in 1983.&lt;ref&gt;{{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}}&lt;/ref&gt; 

=== 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].”&lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/about-us/|title=About Us|date=2024|website=Leabhar Breac|access-date=1 July 2025}}&lt;/ref&gt; 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.&lt;ref name=&quot;:0&quot; /&gt; Leabhar Breac aims to publish Irish-language books that meet “a high literary and artistic standard.”&lt;ref name=&quot;:0&quot; /&gt; Besides the content, Leabhar Breac is known for the typically &quot;superb [thar cionn]&quot; quality of the design and production of the &quot;physical book [leabhar fisiciúil].&quot;&lt;ref name=&quot;:8&quot;&gt;{{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}}&lt;/ref&gt; 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,&lt;ref name=&quot;:1&quot;&gt;{{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}}&lt;/ref&gt; 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&lt;ref&gt;{{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}}&lt;/ref&gt; until 2013.

Besides being its co-director, Ó Scolaí has taken part in this conference as an artist&lt;ref&gt;{{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}}&lt;/ref&gt; and writer&lt;ref name=&quot;:2&quot;&gt;{{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}}&lt;/ref&gt;.

==== 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.&lt;ref name=&quot;:19&quot;&gt;{{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}}&lt;/ref&gt; 

== 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 &quot;named as ‘the best novel since the turn of the Century’ by Comhar.&quot;&lt;ref&gt;{{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}}&lt;/ref&gt;
* ''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.&lt;ref name=&quot;:4&quot;&gt;{{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}}&lt;/ref&gt;
* ''Bódléar'', Leabhar Breac, 2024. The Oireachtas Prize for Literary Fiction, 2024&lt;ref name=&quot;:7&quot; /&gt;; 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.&lt;ref&gt;{{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}}&lt;/ref&gt;

=== 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&lt;sup&gt;th&lt;/sup&gt;-century tale from the [[wikipedia:Fenian_Cycle|Fenian Cycle]].&lt;ref&gt;{{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}}&lt;/ref&gt;
* ''An Ceithearnach Caolriabhach'' (trans., ''The Narrow-Striped Kern''), Leabhar Breac, 2002; a retelling from c. 1500, also illustrated by Darach Ó ScolaÍ.&lt;ref&gt;{{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}}&lt;/ref&gt;
* ''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.&lt;ref name=&quot;:3&quot;&gt;{{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}}&lt;/ref&gt;  &quot;''Táin Bó Cuailnge'' won the Aodán Mac Poilín Memorial Prize 2017.&quot;&lt;ref name=&quot;:19&quot; /&gt;
* ''Deirdre'', Leabhar Breac, 2023, a “picture book for adults” with artist Anastasia Melnykova.&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/deirdre/|title=Deirdre|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt; 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.&lt;ref&gt;{{Cite book|title=A Dictionary of Celtic Mythology|last=MacKillop|first=James|publisher=Oxford University Press|year=2004|isbn=9780198609674|pages=181}}&lt;/ref&gt;
* ''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]].”&lt;ref name=&quot;:4&quot; /&gt;
* ''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'').&lt;ref name=&quot;:5&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/tales-of-wonder/|title=Tales of Wonder|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''Eoghan Béal'', Leabhar Breac, 2025, a picture book for adult readers illustrated by Alé Mercado&lt;ref name=&quot;:5&quot; /&gt;; a retelling of the medieval tale ''[https://ga.wikipedia.org/wiki/Caithr%C3%A9im_Cellaig Cathréim Ceallaigh]'' from ''The Yellow Book of Leacan.''&lt;ref name=&quot;:5&quot; /&gt;

=== 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.”&lt;ref name=&quot;:6&quot;&gt;{{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}}&lt;/ref&gt; [[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.”&lt;ref name=&quot;:9&quot;&gt;{{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}}&lt;/ref&gt;
* ''Dóiteoir na Samhna'' (trans., ''The Halloween Burner''), 2010.&lt;ref name=&quot;:6&quot; /&gt;
* ''Bodach an Chóta Lachna'' (trans., ''The Churl in the Dun Coat''), 2011.&lt;ref&gt;{{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}}&lt;/ref&gt;

The Scéalta Staire (Historical Stories) series&lt;ref name=&quot;:9&quot; /&gt;

* ''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.&lt;ref&gt;{{Cite journal|date=2025-06-19|title=Oireachtas na Gaeilge|url=https://en.wikipedia.org/w/index.php?title=Oireachtas_na_Gaeilge&amp;oldid=1296394643|journal=Wikipedia|language=en}}&lt;/ref&gt;
* 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 &quot;'Twas the Night Before Christmas&quot;), 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.&lt;ref&gt;{{Cite journal|date=2025-07-01|title=Corto Maltese|url=https://en.wikipedia.org/w/index.php?title=Corto_Maltese&amp;oldid=1298285365|journal=Wikipedia|language=en}}&lt;/ref&gt; Ó Scolaí's '''translation of ''Corto Maltese''''' was listed in 2017 among &quot;The 30 Irish books that Irish people love.&quot;&lt;ref&gt;{{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}}&lt;/ref&gt;

* 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 &amp; 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.&lt;ref&gt;{{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}}&lt;/ref&gt; 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''.&lt;ref&gt;{{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}}&lt;/ref&gt; Walter Macken Prize, 2005; BBC Stewart Parker Award, 2006.&lt;ref&gt;{{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}}&lt;/ref&gt;
* ''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.&lt;ref&gt;{{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}}&lt;/ref&gt;
* ''An Braon'' Aníos (trans., ''Rising Damp''), 2006, directed by Ó Scolaí.&lt;ref&gt;{{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}}&lt;/ref&gt; 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.”&lt;ref&gt;{{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}}&lt;/ref&gt;
* ''Craos'' (trans., ''Gluttony''), 2008, directed by Ó Scolaí.&lt;ref name=&quot;:13&quot;&gt;{{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}}&lt;/ref&gt; The third play in the ''Trí Bhraon'' (''Three Drops'') trilogy, it toured to Cork and Belfast.&lt;ref name=&quot;:13&quot; /&gt; 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.”&lt;ref&gt;{{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}}&lt;/ref&gt;
* ''A+E'', 2008, by Ríonach Ní Néill and Darach Ó Scolaí, &quot;dance and music drama,&quot; co-produced by Ciotóg and Salamandar.&lt;ref&gt;{{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}}&lt;/ref&gt;
* ''An tSeanbhróg'' (trans., ''The Old Shoe''), 2009, produced by Salamander&lt;ref&gt;{{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}}&lt;/ref&gt; 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'''.&lt;ref&gt;{{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}}&lt;/ref&gt;

==== Screenplays ====

* ''Cosa Nite'' (trans., ''Washed Feet''), short film, 1998 (dir. Dearbhla Walsh, prod. Ciarán Ó Cofaigh, Rosg); &quot;a prose version of ''Cosa Nite'' was published (Rosg 2000).&quot;&lt;ref name=&quot;:9&quot; /&gt; Nominated for an Irish Film and Television Award.&lt;ref&gt;{{Citation|title=Cosa Nite (Short 1998) - Awards - IMDb|url=https://www.imdb.com/title/tt0191917/awards/|accessdate=2025-08-25|language=en-US}}&lt;/ref&gt;
* ''Na Glúnta'' (trans., ''The Generations''), 2001&lt;ref&gt;{{Cite web|url=https://www.iftn.ie/production/production_companies/production_sub/feature/?act1=record&amp;aid=70&amp;rid=3917&amp;tpl=filmography_dets&amp;only=1&amp;force=1|title=Na Glúnta {{!}} The Irish Film &amp; Television Network|website=www.iftn.ie|access-date=2025-08-25}}&lt;/ref&gt;, co-directors Ciarán Ó Cofaigh &amp; 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.&lt;ref&gt;{{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&amp;apos;Maonlai, Peadar O&amp;apos;Treasaigh, Diarmuid Mac an Adhastair}}&lt;/ref&gt;
* ''Na Cloigne'' [trans., The Heads], 3-episide series, 2010 (dir. Robert Quinn, prod. Ciarán Ó Cofaigh, Rosg), TG4.&lt;ref&gt;{{Cite web|url=https://www.imdb.com/title/tt1607924/|title=Na cloigne|date=2010|website=IMDb|access-date=25 August 2025}}&lt;/ref&gt;
=== 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 &amp; Forgetfulness&quot;]), 2014.
* “Rithim agus Réim” (&quot;Rhythm and Register&quot;), a public lecture in the University College Dublin lecture series “Ó Thrácht go Twitter” (trans., &quot;From Talk to Twitter&quot;), 2014.
* Review of Pádraig Ó Cíobháin’s ''Dréachta Chrích Fodla'', '''Comhar?, ??'''.
* “Na Geilt i mBun an Tí” (trans., &quot;The Madmen in Charge&quot;), a talk at the Merriman Winter School, Comhar April 2012.&lt;ref name=&quot;:18&quot;&gt;{{Cite web|url=http://darachoscolai.ie/beathaisneis.html|title=Darach Ó Scolaí: Beathaisnéis|website=darachoscolai.ie|access-date=2025-09-26}}&lt;/ref&gt;
* 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,”&lt;ref&gt;{{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}}&lt;/ref&gt; with “an unsurpassed richness and precision of language.”&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Ó Cróinín|first=Breandán|date=Summer 2025|title=unknown|journal=The Limerick Leader}}&lt;/ref&gt; “Whimsical, hilarious, and subtly learned” is how Éilis Ní Dhuibhne described his writing.&lt;ref name=&quot;:20&quot; /&gt;   

=== 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.”&lt;ref name=&quot;:12&quot; /&gt; 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].”&lt;ref name=&quot;:11&quot; /&gt;{{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,&lt;blockquote&gt;This then is a novel in search of a plot, a story that attempts to attain a significance that eludes it.&lt;ref&gt;{{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}}&lt;/ref&gt;&lt;/blockquote&gt; 

In the ''Oxford Handbook of Modern Irish Fiction'' Pádraig Ó Siadhail analyzes rather than reviews ''An Cléireach'': &lt;blockquote&gt;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 &quot;Saltair an Easpaig&quot; (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.&lt;ref&gt;{{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}}&lt;/ref&gt; &lt;/blockquote&gt;

Of ''Súil an Daill,'' in ''Nós'', Cathal Seoighe says, &quot;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].&quot;&lt;ref&gt;{{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}}&lt;/ref&gt;

''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.”&lt;ref&gt;{{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}}&lt;/ref&gt; Éilis Ní Dhuibhne in the ''Irish Times'' says,&lt;blockquote&gt;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!&lt;ref name=&quot;:20&quot;&gt;{{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}}&lt;/ref&gt;&lt;/blockquote&gt;

=== 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.&lt;ref name=&quot;:3&quot; /&gt; Gearóid Denvir reviewed ''Táin Bó Cuailnge'' for ''Comhar'':&lt;blockquote&gt;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.]&lt;ref name=&quot;:15&quot;&gt;{{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}}&lt;/ref&gt; &lt;/blockquote&gt;Cathal Poirtéir says, &quot;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.&quot;&lt;ref&gt;{{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}}&lt;/ref&gt;{{rp|47}} Novelist and academic Alan Titley calls Ó Scolaí's &quot;a wonderful gutsy telling&quot; of ''Táin Bó Cuailnge''.&lt;ref&gt;{{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}}&lt;/ref&gt;

==== ''Deirdre'' ====

Marie Whelton, in &quot;Léann Teanga&quot; (&quot;Language Studies&quot;), in the 2024 ''An Reiviú'' says,&lt;blockquote&gt;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.].&lt;ref name=&quot;:16&quot;&gt;{{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}}&lt;/ref&gt;&lt;/blockquote&gt;
=== Works for Young Readers ===
Meadhbh Ní Eadhra said of ''Bodach an Chóta Lachna'' that it was &quot;Beautiful Irish, but easy to understand for young readers.&quot;&lt;ref&gt;Ní Eadhra, Meadhbh. In ''Gaelscéal'', qtd. in &quot;Bodach an Chóta Lachna&quot; https://leabharbreac.com/en/shop/oige-en/7-8/bodach-an-chota-lachna/.&lt;/ref&gt;

== 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].”&lt;ref name=&quot;:18&quot; /&gt;
* 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.&lt;ref name=&quot;:10&quot;&gt;{{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}}&lt;/ref&gt;

* ''An Cléireach'' (''The Clerk'').&lt;ref&gt;{{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}}&lt;/ref&gt;
* ''Táin Bó Cuailnge'', 2018.
* ''Bódléar'', 2025.

==== De Bhaldraithe Award ====
The Gradam de Bhaldraithe is awarded to the best work in translation.&lt;ref name=&quot;:10&quot; /&gt;

* ''Cuairt San Nioclás,'' a translation of Clement Clarke Moore's ''A Visit from St. Nicholas'', or &quot;'Twas the Night Before Christmas.&quot;&lt;ref name=&quot;:10&quot; /&gt;

==== 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&lt;ref&gt;{{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}}&lt;/ref&gt;
* 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/ &lt;nowiki&gt;http://ww&lt;/nowiki&gt;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 (&lt;nowiki&gt;https://www.dib.ie&lt;/nowiki&gt;) 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 ===

* &quot;So here are the books that Irish people love the most! [Mar sin seo iad na leabhair is gile leis na Gaeil!]&quot; — &quot;32. An Cléireach – Darach Ó Scolaí (2)&quot; [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]&lt;ref name=&quot;:14&quot;&gt;{{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}}&lt;/ref&gt;
* &quot;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].&quot; &quot;Corto Maltese – Hugo Pratt (aistrithe ag Darach Ó Scolaí)&quot;&lt;ref name=&quot;:14&quot; /&gt;
* &quot;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.]&quot;&lt;ref name=&quot;:17&quot;&gt;{{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}}&lt;/ref&gt;

=== 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.’&quot;

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 &lt;ref name=&quot;:15&quot; /&gt;]

“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 &amp; 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&lt;ref&gt;{{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}}&lt;/ref&gt;:

*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 &quot;First Readers&quot; (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).&lt;ref&gt;{{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}}&lt;/ref&gt; 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.”&lt;ref name=&quot;:9&quot; /&gt;

== References ==
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    <title>User:Dc.samizdat/Real Euclidean four-dimensional space R⁴</title>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="124212" sha1="9cmystr80kvu0rah7bh0zk6wnnb69mt" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth 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 &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth 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 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 velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical 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 has thickness not only in the third dimension, but in the fourth dimension as well. The central ball-like region is a 4-ball.  

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, 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 it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we observe 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 center point, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories 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 occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements, the compound of a simple rotation and a 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. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. 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 finite 3-space 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 on 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 objects we observe did not, and lie outside our big-bang's 3-sphere of outflying matter, or even inside its 3-sphere, below its surface. For all observers, the conjectured big-bang origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured 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 &lt;math&gt;c&lt;/math&gt;. 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 the 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.   

The enclosing surface of a spherical region of 4-space of any size is itself 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 we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-space, in some 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 containing 3-sphere is one of our galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our solar system lies on one of the concentric 3-spheres of our 4-ball galaxy. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.  

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their internal geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball.

Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two 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 &lt;math&gt;c&lt;/math&gt; in 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 &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt; in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and 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}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first described by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

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

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

...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.

Euclidean relativity is not even a fringe theory; no physicists 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 &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, 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. 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:

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

While this description should be understood as 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 (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), reflection (&lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt;) and translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) are just what they are in three-dimensional space, but double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space to rotate in.

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

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

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

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...

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

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; 

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

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

...

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

...this is probably misplaced here and should not interrupt the discussion at this point:
...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

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

...cite Jesper Goransson's very concise paper

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

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

...

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

...

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

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

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

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

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

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

...

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

...

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

== Distribution of stars in our galaxy ==

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

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

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

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

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

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

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

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

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

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

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

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

== Light and Mass are Reflection and Rotation ==

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

...

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

Light is ....

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

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

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

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

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

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

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

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

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

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

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

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

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

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|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 &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>9cmystr80kvu0rah7bh0zk6wnnb69mt</sha1>
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      <id>2829735</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="124247" sha1="1z6d734fko6tqwr0s9kne6922rpt2e4" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth 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 &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth 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 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 velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical 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 has thickness not only in the third dimension, but in the fourth dimension as well. The central ball-like region is a 4-ball.  

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, 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 it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we observe 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 center point, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories 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 occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements in 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. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. 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 finite 3-space 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 on 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 objects we observe did not, and lie outside our big-bang's 3-sphere of outflying matter, or even inside its 3-sphere, below its surface. For all observers, the conjectured big-bang origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured 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 &lt;math&gt;c&lt;/math&gt;. 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 the 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.   

The enclosing surface of a spherical region of 4-space of any size is itself 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 we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-space, in some 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 containing 3-sphere is one of our galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our solar system lies on one of the concentric 3-spheres of our 4-ball galaxy. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.  

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their internal geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball.

Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two 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 &lt;math&gt;c&lt;/math&gt; in 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 &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt; in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and 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}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first described by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

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

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

...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.

Euclidean relativity is not even a fringe theory; no physicists 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 &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, 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. 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:

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

While this description should be understood as 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 (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), reflection (&lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt;) and translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) are just what they are in three-dimensional space, but double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space to rotate in.

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

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

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

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...

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

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; 

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

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

...

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

...this is probably misplaced here and should not interrupt the discussion at this point:
...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

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

...cite Jesper Goransson's very concise paper

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

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

...

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

...

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

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

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

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

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

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

...

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

...

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

== Distribution of stars in our galaxy ==

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

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

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

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

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

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

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

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

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

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

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

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

== Light and Mass are Reflection and Rotation ==

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

...

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

Light is ....

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

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

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

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

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

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

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

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

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

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

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

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

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

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|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 &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2829736</id>
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      <timestamp>2026-08-30T16:18:00Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="124257" sha1="0mezj5w05c1qrs4zn0jiibdgxd87gcj" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth 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 &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth 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 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 velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical 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 has thickness not only in the third dimension, but in the fourth dimension as well. The central ball-like region is a 4-ball.  

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, 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 it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we observe 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 center point, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories 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 occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements in 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 motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. 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 finite 3-space 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 on 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 objects we observe did not, and lie outside our big-bang's 3-sphere of outflying matter, or even inside its 3-sphere, below its surface. For all observers, the conjectured big-bang origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured 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 &lt;math&gt;c&lt;/math&gt;. 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 the 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.   

The enclosing surface of a spherical region of 4-space of any size is itself 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 we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-space, in some 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 containing 3-sphere is one of our galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our solar system lies on one of the concentric 3-spheres of our 4-ball galaxy. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.  

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their internal geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball.

Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two 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 &lt;math&gt;c&lt;/math&gt; in 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 &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt; in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and 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}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first described by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

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

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

...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.

Euclidean relativity is not even a fringe theory; no physicists 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 &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, 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. 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:

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

While this description should be understood as 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 (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), reflection (&lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt;) and translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) are just what they are in three-dimensional space, but double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space to rotate in.

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

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

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

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...

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

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; 

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

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

...

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

...this is probably misplaced here and should not interrupt the discussion at this point:
...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

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

...cite Jesper Goransson's very concise paper

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

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

...

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

...

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

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

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

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

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

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

...

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

...

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

== Distribution of stars in our galaxy ==

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

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

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

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

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

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

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

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

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

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

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

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

== Light and Mass are Reflection and Rotation ==

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

...

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

Light is ....

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

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

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

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

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

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

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

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

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

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

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

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

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

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|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 &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <id>2829737</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="124407" sha1="ai9y139aju8tuvadgitmssn31wdhc7r" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth 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 &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth 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 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 velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical 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 has thickness not only in the third dimension, but in the fourth dimension as well. The central ball-like region is a 4-ball.  

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, 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 it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we observe 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 center point, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories 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 occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements in 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 motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. 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 finite 3-space 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 on the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang as us long ago. Possibly some of the objects we observe did not, and lie outside our big-bang's 3-sphere of outflying matter, or even inside its 3-sphere, below its surface. In any case we should not assume that all objects in the 4-space universe lie on the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang origin point 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 &lt;math&gt;c&lt;/math&gt;. 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 the 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.   
The enclosing surface of a spherical region of 4-space of any size is itself 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 we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-space, in some 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 containing 3-sphere is one of our galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our solar system lies on one of the concentric 3-spheres of our 4-ball galaxy. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.  

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their internal geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball.

Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two 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 &lt;math&gt;c&lt;/math&gt; in 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 &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt; in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and 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}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first described by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

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

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

...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.

Euclidean relativity is not even a fringe theory; no physicists 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 &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, 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. 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:

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

While this description should be understood as 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 (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), reflection (&lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt;) and translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) are just what they are in three-dimensional space, but double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space to rotate in.

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

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

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

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...

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

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; 

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

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

...

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

...this is probably misplaced here and should not interrupt the discussion at this point:
...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

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

...cite Jesper Goransson's very concise paper

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

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

...

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

...

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

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

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

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

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

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

...

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

...

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

== Distribution of stars in our galaxy ==

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

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

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

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

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

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

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

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

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

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

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

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

== Light and Mass are Reflection and Rotation ==

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

...

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

Light is ....

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

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

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

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

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

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

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

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

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

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

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

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

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

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|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 &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>ai9y139aju8tuvadgitmssn31wdhc7r</sha1>
    </revision>
    <revision>
      <id>2829738</id>
      <parentid>2829737</parentid>
      <timestamp>2026-08-30T16:36:37Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="124438" sha1="n262450nacec1soudiy8oxy2ts2k25o" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth 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 &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth 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 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 velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical 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 has thickness not only in the third dimension, but in the fourth dimension as well. The central ball-like region is a 4-ball.  

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, 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 it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we observe 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 center point, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories 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 occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements in 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 motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. 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 finite 3-space 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 on the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang as us long ago. Possibly some of the objects we observe did not, and lie outside our big-bang's 3-sphere of outflying matter, or even inside its 3-sphere, below its surface. In any case we should not assume that all objects in the 4-space universe lie on the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang origin point 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 &lt;math&gt;c&lt;/math&gt;. 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 the 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.   
The enclosing surface of a spherical region of 4-space of any size is itself 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 we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-space, in some 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 containing 3-sphere is one of our galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our solar system lies on one of the concentric 3-spheres of our 4-ball galaxy. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.  

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their internal geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball.

Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two 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 &lt;math&gt;c&lt;/math&gt; in 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 &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt; in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and 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}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first described by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

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

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

...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.

Euclidean relativity is not even a fringe theory; no physicists 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 &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, 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. 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:

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

While this description should be understood as 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 (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), reflection (&lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt;) and translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) are just what they are in three-dimensional space, but double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space to rotate in.

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

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

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

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...

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

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; 

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

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

...

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

...this is probably misplaced here and should not interrupt the discussion at this point:
...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

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

...cite Jesper Goransson's very concise paper

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

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

...

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

...

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

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

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

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

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

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

...

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

...

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

== Distribution of stars in our galaxy ==

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

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

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

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

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

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

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

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

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

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

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

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

== Light and Mass are Reflection and Rotation ==

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

...

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

Light is ....

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

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

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

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

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

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

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

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

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

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

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

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

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

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|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 &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <timestamp>2026-08-30T16:39:16Z</timestamp>
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        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="124432" sha1="dw5zvez6d39jos9yvi7h1u25ji848c7" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth 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 &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth 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 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 velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical 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 has thickness not only in the third dimension, but in the fourth dimension as well. The central ball-like region is a 4-ball.  

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, 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 it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we observe 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 center point, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories 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 occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements in 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 motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. 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 finite 3-space 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 on 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 objects we observe did not, and lie outside our big-bang's 3-sphere of outflying matter, or even inside its 3-sphere, below its surface. In any case we should not assume that all objects in the 4-space universe lie on the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang origin point 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 &lt;math&gt;c&lt;/math&gt;. 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 the 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.   
The enclosing surface of a spherical region of 4-space of any size is itself 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 we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-space, in some 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 containing 3-sphere is one of our galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our solar system lies on one of the concentric 3-spheres of our 4-ball galaxy. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.  

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their internal geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball.

Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two 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 &lt;math&gt;c&lt;/math&gt; in 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 &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt; in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and 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}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first described by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

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

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

...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.

Euclidean relativity is not even a fringe theory; no physicists 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 &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, 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. 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:

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

While this description should be understood as 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 (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), reflection (&lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt;) and translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) are just what they are in three-dimensional space, but double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space to rotate in.

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

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

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

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...

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

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; 

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

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

...

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

...this is probably misplaced here and should not interrupt the discussion at this point:
...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

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

...cite Jesper Goransson's very concise paper

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

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

...

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

...

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

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

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

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

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

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

...

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

...

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

== Distribution of stars in our galaxy ==

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

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

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

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

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

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

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

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

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

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

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

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

== Light and Mass are Reflection and Rotation ==

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

...

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

Light is ....

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

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

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

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

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

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

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

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

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

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

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

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

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

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|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 &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <id>2829741</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="124346" sha1="j31a4vancxb6202icjsoi1zelfcrkow" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth 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 &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth 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 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 velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical 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 has thickness not only in the third dimension, but in the fourth dimension as well. The central ball-like region is a 4-ball.  

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, 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 it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we observe 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 center point, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories 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 occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements in 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 motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. 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 finite 3-space 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 on 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 objects we observe did not, and lie outside our big-bang's 3-sphere of outflying matter, or even inside its 3-sphere, below its surface. In any case we should not assume that all objects in the 4-space universe lie on the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang origin point 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 &lt;math&gt;c&lt;/math&gt;. 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.   
The enclosing surface of a spherical region of 4-space of any size is itself 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 we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 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 &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their internal geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball.

Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two 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 &lt;math&gt;c&lt;/math&gt; 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 &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion 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 &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and 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}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first described by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

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

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

...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.

Euclidean relativity is not even a fringe theory; no physicists 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 &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, 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. 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:

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

While this description should be understood as 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 (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), reflection (&lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt;) and translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) are just what they are in three-dimensional space, but double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space to rotate in.

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

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

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

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...

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

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; 

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

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

...

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

...this is probably misplaced here and should not interrupt the discussion at this point:
...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

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

...cite Jesper Goransson's very concise paper

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

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

...

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

...

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

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

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

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

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

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

...

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

...

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

== Distribution of stars in our galaxy ==

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

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

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

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

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

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

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

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

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

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

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

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

== Light and Mass are Reflection and Rotation ==

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

...

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

Light is ....

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

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

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

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

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

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

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

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

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

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

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

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

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

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|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 &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <id>2829742</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="124346" sha1="4tfxk6ouydijjxyl8b1n8x0fverqpe8" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth 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 &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth 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 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 velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical 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 has thickness not only in the third dimension, but in the fourth dimension as well. The central ball-like region is a 4-ball.  

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, 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 it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we observe 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 center point, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories 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 occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements in 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 motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. 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 finite 3-space 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 on 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 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 should not 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 origin point 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 &lt;math&gt;c&lt;/math&gt;. 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.   
The enclosing surface of a spherical region of 4-space of any size is itself 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 we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 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 &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their internal geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball.

Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two 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 &lt;math&gt;c&lt;/math&gt; 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 &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion 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 &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and 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}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first described by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

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

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

...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.

Euclidean relativity is not even a fringe theory; no physicists 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 &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, 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. 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:

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

While this description should be understood as 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 (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), reflection (&lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt;) and translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) are just what they are in three-dimensional space, but double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space to rotate in.

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

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

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

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...

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

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; 

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

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

...

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

...this is probably misplaced here and should not interrupt the discussion at this point:
...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

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

...cite Jesper Goransson's very concise paper

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

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

...

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

...

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

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

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

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

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

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

...

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

...

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

== Distribution of stars in our galaxy ==

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

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

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

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

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

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

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

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

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

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

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

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

== Light and Mass are Reflection and Rotation ==

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

...

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

Light is ....

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

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

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

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

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

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

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

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

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

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

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

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

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

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|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 &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <id>2829743</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="124366" sha1="3unwchtbphax979s97ya5rjlqte9dcx" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth 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 &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth 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 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 velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical 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 has thickness not only in the third dimension, but in the fourth dimension as well. The central ball-like region is a 4-ball.  

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, 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 it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we observe 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 center point, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories 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 occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements in 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 motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. 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 finite 3-space 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 on 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 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 should not 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 origin point 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 &lt;math&gt;c&lt;/math&gt;. 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.   
The enclosing surface of a spherical region of 4-space of any size is itself 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 we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, rounded by gravity, 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 tiny patch of a filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 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 &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their internal geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball.

Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two 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 &lt;math&gt;c&lt;/math&gt; 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 &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion 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 &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and 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}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first described by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

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

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

...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.

Euclidean relativity is not even a fringe theory; no physicists 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 &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, 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. 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:

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

While this description should be understood as 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 (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), reflection (&lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt;) and translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) are just what they are in three-dimensional space, but double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space to rotate in.

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

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

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

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...

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

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; 

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

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

...

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

...this is probably misplaced here and should not interrupt the discussion at this point:
...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

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

...cite Jesper Goransson's very concise paper

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

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

...

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

...

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

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

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

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

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

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

...

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

...

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

== Distribution of stars in our galaxy ==

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

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

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

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

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

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

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

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

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

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

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

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

== Light and Mass are Reflection and Rotation ==

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

...

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

Light is ....

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

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

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

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

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

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

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

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

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

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

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

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

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

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|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 &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>3unwchtbphax979s97ya5rjlqte9dcx</sha1>
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    <revision>
      <id>2829744</id>
      <parentid>2829743</parentid>
      <timestamp>2026-08-30T17:35:12Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="124374" sha1="rhehx0877s6kw51rfgclz21idau6mb0" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth 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 &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth 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 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 velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical 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 has thickness not only in the third dimension, but in the fourth dimension as well. The central ball-like region is a 4-ball.  

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, 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 it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we observe 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 center point, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories 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 occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements in 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 motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. 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 finite 3-space 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 on 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 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 should not 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 origin point 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 &lt;math&gt;c&lt;/math&gt;. 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.   
The enclosing surface of a spherical region of 4-space of any size is itself 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 we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, rounded by gravity, 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 tiny patch of a filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold 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 &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their internal geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball.

Every such moving 3-dimensional surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two 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 &lt;math&gt;c&lt;/math&gt; 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 &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion 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 &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and 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}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first described by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

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

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

...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.

Euclidean relativity is not even a fringe theory; no physicists 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 &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, 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. 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:

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

While this description should be understood as 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 (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), reflection (&lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt;) and translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) are just what they are in three-dimensional space, but double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space to rotate in.

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

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

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

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...

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

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; 

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

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

...

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

...this is probably misplaced here and should not interrupt the discussion at this point:
...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

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

...cite Jesper Goransson's very concise paper

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

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

...

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

...

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

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

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

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

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

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

...

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

...

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

== Distribution of stars in our galaxy ==

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

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

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

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

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

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

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

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

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

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

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

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

== Light and Mass are Reflection and Rotation ==

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

...

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

Light is ....

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

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

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

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

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

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

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

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

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

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

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

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

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

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|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 &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <timestamp>2026-08-30T17:37:57Z</timestamp>
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        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="124360" sha1="bkrx47ff9zdag6qlkeyq106lbvc1so1" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth 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 &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth 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 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 velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical 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 has thickness not only in the third dimension, but in the fourth dimension as well. The central ball-like region is a 4-ball.  

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, 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 it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we observe 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 center point, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories 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 occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements in 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 motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. 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 finite 3-space 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 on 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 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 should not 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 origin point 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 &lt;math&gt;c&lt;/math&gt;. 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.   
The enclosing surface of a spherical region of 4-space of any size is itself 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 we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, rounded by gravity, 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 tiny patch of a filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold 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 &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their internal geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball.

Every such moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two 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 &lt;math&gt;c&lt;/math&gt; 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 &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion 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 &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and 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}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first described by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

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

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

...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.

Euclidean relativity is not even a fringe theory; no physicists 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 &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, 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. 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:

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

While this description should be understood as 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 (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), reflection (&lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt;) and translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) are just what they are in three-dimensional space, but double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space to rotate in.

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

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

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

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...

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

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; 

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

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

...

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

...this is probably misplaced here and should not interrupt the discussion at this point:
...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

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

...cite Jesper Goransson's very concise paper

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

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

...

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

...

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

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

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

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

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

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

...

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

...

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

== Distribution of stars in our galaxy ==

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

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

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

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

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

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

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

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

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

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

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

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

== Light and Mass are Reflection and Rotation ==

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

...

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

Light is ....

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

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

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

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

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

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

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

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

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

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

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

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

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

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|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 &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <id>2829746</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="124362" sha1="3n3r2rprxttr8oio8iypg3j9m3s3ngp" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth 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 &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth 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 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 velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical 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 has thickness not only in the third dimension, but in the fourth dimension as well. The central ball-like region is a 4-ball.  

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, 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 it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we observe 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 center point, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories 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 occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements in 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 motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. 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 finite 3-space 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 on 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 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 should not 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 origin point 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 &lt;math&gt;c&lt;/math&gt;. 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.   
The enclosing surface of a spherical region of 4-space of any size is itself 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 we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, rounded by gravity, 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 tiny patch of a filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold 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 &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their internal geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball.

Every such moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two 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 &lt;math&gt;c&lt;/math&gt; 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 &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion 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 &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and 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}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first described by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

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

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

...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.

Euclidean relativity is not even a fringe theory; no physicists 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 &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, 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. 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:

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

While this description should be understood as 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 (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), reflection (&lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt;) and translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) are just what they are in three-dimensional space, but double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space to rotate in.

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

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

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

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...

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

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; 

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

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

...

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

...this is probably misplaced here and should not interrupt the discussion at this point:
...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

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

...cite Jesper Goransson's very concise paper

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

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

...

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

...

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

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

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

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

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

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

...

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

...

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

== Distribution of stars in our galaxy ==

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

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

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

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

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

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

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

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

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

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

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

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

== Light and Mass are Reflection and Rotation ==

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

...

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

Light is ....

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

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

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

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

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

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

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

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

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

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

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

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

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

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|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 &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <id>2829767</id>
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      <timestamp>2026-08-30T19:29:53Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="124369" sha1="j197i8g64z5p22fc2368h9uv80a1fs3" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth 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 &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth 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 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 velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical 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 has thickness not only in the third dimension, but in the fourth dimension as well. The central ball-like region is a 4-ball.  

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, 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 it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we observe 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 center point, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories 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 occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements in 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 motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. 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 on 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 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 should not 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 origin point 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 &lt;math&gt;c&lt;/math&gt;. 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.   
The enclosing surface of a spherical region of 4-space of any size is itself 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 we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 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 tiny patch of 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 entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of rotating objects through Euclidean space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold 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 &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their internal geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball.

Every such moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two 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 &lt;math&gt;c&lt;/math&gt; 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 &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion 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 &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and 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}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first described by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

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

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

...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.

Euclidean relativity is not even a fringe theory; no physicists 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 &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, 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. 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:

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

While this description should be understood as 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 (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), reflection (&lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt;) and translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) are just what they are in three-dimensional space, but double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space to rotate in.

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

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

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

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...

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

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; 

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

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

...

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

...this is probably misplaced here and should not interrupt the discussion at this point:
...These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), since one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

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

...cite Jesper Goransson's very concise paper

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

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

...

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

...

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

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

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

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

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

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

...

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

...

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

== Distribution of stars in our galaxy ==

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

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

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

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

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

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

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

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

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

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

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

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

== Light and Mass are Reflection and Rotation ==

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

...

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

Light is ....

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

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

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

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

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

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

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

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

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

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

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

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

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

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|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 &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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{{MECR3|1=https://youtu.be/8CafEcnbuHk}}
__TOC__

==Overview==
{{RoundBoxTop|theme=2}}'''Case study'''
[[File:MUGNIFYING GLASS ICON BY V.V ICONS.jpg|50px]]{{clear}}
[[File:Workforce data summary.jpg|thumb|'''Figure 1.''' Fictional summary of the monthly workforce data report.|right|250px]]
[[File:Employee Survey Data Summary.jpg|thumb|'''Figure 2.''' Fictional snapshot of the 2023 employee survey results.|right|250px]] 

Sarah is a Senior Executive Manager in a fast-paced government role, having quickly risen through the ranks due to her technical policy expertise. Currently, she leads a diverse team of 51 employees who are working on a critical new policy proposal with tight deadlines.  

Each month, Sarah reviews her workforce data report (see Figure 1), which reflects staff capacity, flexibility, leave, diversity, and mobility. After reviewing her most recent report she noticed that the unscheduled absence rate and separations were significantly higher than previous months. One thing Sarah has observed is that tensions are running high in the team. Due to other priorities, Sarah has chosen to ignore the issue, hoping her directors will support and manage their teams more effectively. During a leadership meeting, directors expressed frustration with Sarah's leadership, informing her that her staff members were feeling overwhelmed by the workload. 

In addition, Sarah received the annual Employee Survey results (see Figure 2) which are used to collect confidential and opinion information on important issues in the workplace. The survey results indicated that staff have a negative perception of their immediate supervisor and Sarah's leadership and effectiveness. Upon reflection, Sarah recognises that she has not demonstrated effective leadership behaviors to her team, which include recognising emotions, responding empathetically, and creating a safe space for team members to express their feelings through open dialogue. In essence, Sarah recognises the need to embody effective leadership behaviours, but what does it take?

{{RoundBoxBottom}}

In today's dynamic and ever-evolving organisational landscape, effective leadership has become an important ingredient for success. Leaders, such as Sarah, are now confronted with a host of complex challenges that extend far beyond conventional notions of authority and decision-making. As organisations grapple with economic uncertainty, high rates of employee turnover, and the relentless pace of change, the role of leaders has transformed into a multifaceted and demanding endeavor. This book chapter investigates the intricate world of effective leadership, using Sarah's journey as a case study. In doing so, the aim is to uncover the essential behaviours, skills, qualities (traits) and strategies that can empower leaders to not only survive but thrive amidst the challenges of contemporary organisational landscapes. This exploration seeks to provide invaluable insights and guidance for leaders, aspiring leaders, and anyone interested in the art and science of effective leadership.

{{RoundBoxTop|theme=3}}'''Focus questions:'''
* What is effective leadership?
* What psychological theories assist with understanding effective leadership?
* What behaviours, skills and traits are required to be an effective leader?
* What strategies can develop effective leaders?
{{RoundBoxBottom}}

== What is effective leadership? ==
[[Motivation and emotion/Book/2018/Emotional intelligence and leadership effectiveness|Effective leadership]] is a multifaceted concept that involves the ability of a leader to guide, inspire, and influence individuals or groups toward the achievement of common goals or objectives, involving a combination of behaviours, skills and traits (ChatGPT August 3 Version. What is effective leadership?) (Horner, 1997).{{expand}}

==What psychological theories assist with understanding effective leadership?==
To assist with understanding effective leadership it is important to consider what psychological theory best explains this phenomenon. There are several possible theories which may be applied, however, the focus will be on the following psychological theories:

=== Emotional intelligence ===
[[File:Model of emotional intelligence.png|thumb|Figure 3: Salovey and Mayer's ability-based EI model.|315x315px]]

Salovey and Mayer (1990) formally introduced the theory of [[wikipedia:Emotional_intelligence|emotional intelligence]] (EI) defining it as {{quote|the ability to monitor one’s own and others’ feelings and emotions, to discriminate among them and to use this information to guide one’s thinking and actions (p. 189).|quote}}
The focus of Salovey and Mayer's EI theory was on an ability-based model (see Figure 3). The model is underpinned by the [[wikipedia:Mayer-Salovey-Caruso_Emotional_Intelligence_Test|Mayer-Salovey-Caruso Emotional Intelligence Test]] (MSCEIT), which measures an individual's ability on the four branches of emotional intelligence (Brackett, Rivers, &amp; Salovey, 2011){{comment|What are the 4 branches?}}.

In 1995, Goleman further developed the theory by introducing a mixed model, encompassing ability, personality traits and competencies, such as optimism and self-esteem (Brackett, Rivers, &amp; Salovey, 2011). Goleman's (1995) emotional intelligence model has five domains:  

# [https://www.verywellmind.com/what-is-self-awareness-2795023 Self-awareness] - an individual recognising and understanding their own emotions as they happen, as well as their strengths, weaknesses, values and goals.
# [[wikipedia:Emotional_self-regulation|Self-regulation]] - the ability for an individual to manage their emotions.
# [[wikipedia:Social_skills|Social skills]] - the ability to manage the emotions of others.
# [[wikipedia:Empathy|Empathy]] - recognising the emotions in others, in essence, the ability to walk in someone else's shoes.
# [[wikipedia:Motivation|Motivation]] - the inner or external factors that drive an individual to achieve challenging short- or long-term goals.

The purpose of these two EI models is to offer a framework for examining and understanding how individuals process emotional information and how it may be applied to raise awareness of behaviours to improve personal and professional relationships. According to Salovey and Mayer (1990), individuals possessing EI skills tend to exhibit greater creativity and adaptability when it comes to problem solving, which is an essential leadership skill. Additionally, they are more inclined to incorporate emotional factors into their decision-making process when evaluating these solutions. This approach promotes behaviour that is thoughtful and considerate, both in relation to one's own internal experiences and those of others (Salovey &amp; Mayer,1990).    

'''Application'''    

With application to the leadership case study, Sarah’s technical policy expertise secured her initial promotion, however this may not be sufficient for securing her next one. If Sarah’s ambition is to assume a leadership position, she must also take into account the emotional aspect. By taking this into account she will be able to coach her team, manage stress, provide feedback and collaborate with others.    

Research findings indicate there is a relationship between EI and several components of [[wikipedia:Transformational_leadership|transformational leadership]], suggesting that it may be an important component of effective leadership (Palmer et al., 2001). In particular, emotional intelligence plays a role for how successful leaders observe, engage with, and nurture their employees' emotional well-being within the workplace (Palmer et al., 2001).

=== Self-determination theory ===
Developed by Deci and Ryan, [[wikipedia:Self-determination_theory|Self-Determination Theory]] (SDT) is a psychological framework that focuses on motivation and human personality, which is centred on an individual's personal growth and their fundamental psychological needs of autonomy, competence and relatedness. From an organisational perspective, this framework also provides leaders with a basis for how to effectively motivate employees (Forner et al., 2020). Summarised from research by Forner, et al. (2020) and Fowler (2018) these psychological needs from a leadership perspective are as follows:
[[File:SelfDeterminationTheory.png|thumb|315x315px|Figure 4: Model of self-determination theory]]

* '''Autonomy''' refers to an employees’ desire to feel they have choice in their role, that they are the source of making their own decisions, and they are able to freely express ideas and decide how the work is done.
* '''Competence''' refers to employees’ need to feel effective, demonstrating skill over time, and feeling like they are successful at their job.
* '''Relatedness''' refers to an employees’ need to feel connected, accepted and to experience a sense of belonging and feeling by others. Employees’ also need to feel they are contributing to something greater, for example, organisational goals.

Leaders who facilitate the fulfillment of these basic three psychological needs foster a state of quality motivation, in which employees genuinely embrace and willingly engage in their job responsibilities (Forner et al., 2020).

'''Application''' 

With application to the leadership case study, it appears that Sarah and her directors may not be meeting the basic psychological needs of their employees. For example:

* Autonomy - Not providing opportunities for team members to express their ideas and suggestions about the work being undertaken on the NPP, which is causing frustration.
* Competence - Not building team members skills, capabilities and self confidence in a safe and supportive environment.
* Relatedness - Social interactions and interpersonal relationships between leaders and team members are strained, due to frustrations about the work.

=== Social exchange theory ===
[[wikipedia:Social_exchange_theory|Social exchange theory]] (SET) is a psychological and sociological theory based on behaviourism, that can be applied to understand workplace behaviour. In the workplace, the fundamental premise of SET is, employees participate in social relationships and interactions guided by the principle of reciprocity. Research indicates, there is an &quot;anticipation of mutual give-and-take&quot; between leaders and employees (Thomas &amp; Gupta, 2021). When an employee perceives that their contributions are not being reciprocated by leaders, it may have an impact on their job performance. One construct which aligns with the principles of SET and explores the outcomes of job performance is bottom-line mentality. 

=== Bottom-line mentality ===
[[Motivation and emotion/Book/2024/Bottom-line mentality and motivation|Bottom-line mentality]] (BLM) is a construct that places a strong emphasis on results, which is described as “one-dimensional thinking that revolves around securing bottom-line outcomes to the neglect of competing priorities” (Greenbaum et al., 2012, p. 344). In a systematic literature review by Greenbaum et al (2023), their findings suggest that BLM may be used as an influencing strategy by people in positions of authority. The findings also confirm that BLM is associated with favourable (e.g., financial and employee performance) and unfavourable (e.g., employee morale and regulatory compliance) outcomes. Essentially, the findings suggest that leaders should exercise substantial caution before endorsing a BLM and consider using other approaches to avoid this mindset.

'''Application'''

With application to the case study, it could be inferred that Sarah unconsciously has bottom line mentality. It appears Sarah is focused on delivering the NPP above all else, which is at the neglect of her team and their needs. While Sarah may not identify herself as having a bottom-line mentality, her employees, however, may perceive this as her leadership style.

=== Emerging themes ===
'''Emotional agility''' 

Dr. Susan David (2016), a psychologist and author, popularised the concept of emotional agility, which refers to the ability of being aware of your emotions and adapting to challenging situations with mindfulness, emotional intelligence, and resilience. It also involves accepting your emotions without judgment, learning to sit in discomfort, and then choosing how to respond to them in a way that aligns with your values and goals (David 2016). 

David (2016) highlights the importance of emotional agility in the workplace, challenging the prevailing wisdom that negative thoughts and emotions should have no place at work and arguing that everyone experiences criticism, doubt, and fear in their thoughts and feelings. To develop emotional agility David suggests following this four-step process, which has been adapted from Acceptance and Commitment Therapy:

# Recognise your patterns: Identify when you're caught up in negative thoughts and emotions.
# Label your thoughts and emotions: Objectively label your thoughts and feelings to create distance from them.
# Accept them: Embrace your thoughts and emotions with an open attitude and compassion.
# Act on your values: Make choices aligned with your core values, even in the face of difficult thoughts and emotions.

David (2016) emphasises that developing emotional agility is a long-term process and can lead to improved well-being and better job performance in a complex, fast-changing work environment.

An effective leader who can serve as a role model, providing [[wikipedia:Psychological_safety|psychological safety]], for the team plays an important role in ensuring an organisations{{gr}} overall success, particularly during challenging periods. Throughout the pandemic, agile leaders proved to be valuable assets to their organisations. The organisations that thrived were those with dedicated and adaptable leaders who were willing and able to pivot for the benefit of the organisation.{{fact}}

{{robelbox|theme=13|title=In review}}  
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
{| class=&quot;wikitable&quot; 
|-
|
{| 
|-
|'''Checking understanding'''
|-                
| 1. What is emotional intelligence, and why is it important in personal and professional life?
|-
| 2. In what ways can an understanding of SDT be applied in workplace settings to enhance employee motivation and job satisfaction?
|-
| 3. What is the main factor that contributes to a bottom-line mentality within an organisation?
|}
'''Critical thinking'''
{|
| As an executive leadership coach and mentor, you’re working with Sarah to become a more effective leader in her government role. Firstly, how would you address the current challenges and negative survey results? What steps would you suggest Sarah takes in developing her emotional intelligence for addressing the issues within her team?
|}
|}
&lt;/div&gt;
{{Robelbox/close}}

== What behaviours, skills and traits are required to be an effective leader? ==
{{RoundBoxTop|theme=2}}'''Case study'''
{|
|+ '''Jacinda Ardern, Prime Minister of New Zealand (2017 - 2023): Leadership during crisis'''
|-
|-
| | [[File:NZ_PM_Jacinda_Ardern_-_Kirk_HargreavesCCC.jpg|200x400px|thumb|Figure 5. Jacinda Ardern showing the utmost empathy while visitng members of the Muslim community at the Phillipstown Community Centre.]] || Jacinda Ardern's leadership during the Christchurch Mosque shootings (see, Figure 5), Whakaari White Island volcano eruption and the COVID-19 pandemic exemplifies the qualities of empathy, effective communication, support, decisiveness, and inclusivity. Her leadership approach not only brought New Zealand through difficult times but also serves as a model for leaders worldwide. Through her actions and gun law policy, Ardern made a lasting impact on her country and the global community, emphasising the importance of compassion and collaboration in times of adversity (see, Figure 6). || [[File:Twitter Example - Jacinda Ardern.png|300x300px|thumb|Figure 6. Ardern's social media was inundated with messages, similar to this one, praising her leadership and compassion.]]
|}
{{RoundBoxBottom}}  

As Prime Minister of New Zealand, [[wikipedia:Jacinda_Ardern|Jacinda Ardern]] showed effective leadership during her tenure, demonstrating a combination of behaviours, skills and traits to inspire and guide her country through a time of crisis. In this next section, we will delve into the research on leadership behaviours, skills and traits, as well as identify some of the important elements of what it takes to be an effective leader, like Jacinda.

=== Behaviours ===
What are leadership behaviours? Effective leadership behaviours are qualities and actions that empower leaders to inspire, influence, and motivate teams (Yukl et al., 2019), thereby improving their effectiveness in achieving work outcomes. These behaviours have been identified by research undertaken by Yukl (1999) {{where}} which generated findings that produced a hierarchical taxonomy comprising of four meta-categories and 15 specific effective leadership behaviours (see Table 1). Leaders can develop behaviours, such as clarifying, supporting, envisioning change and networking, to boost the efficiency and productivity of their team members. Additionally, relations-orientated leadership behaviours contribute significantly to job satisfaction, in particular supporting, developing and empowering employees (Yukl et al., 2019).
 
{| class=&quot;wikitable&quot;
|+
Table 1: Hierarchical taxonomy of leadership behaviours (Yukl, 1999; Yukl et al., 2019)
|'''Meta-categories'''
|'''Specific behaviours'''
|'''Primary objective'''
|-
|{{RoundBoxTop|theme=2}}'''Task-orientated'''{{RoundBoxBottom}}  
|Clarifying
Planning

Monitoring operations

Problem solving
|The '''primary objective''' is to accomplish work in an efficient and 
reliable way.

''Related significantly to managerial effectiveness.''
|-
|{{RoundBoxTop|theme=3}}'''Relations-orientated'''{{RoundBoxBottom}} 
|Supporting
Developing
Recognising

Empowering
|The '''primary objective''' is to increase the quality of human resources.
''Related significantly to managerial effectiveness and job satisfaction.'' 
|-
|{{RoundBoxTop|theme=4}}'''Change-orientated'''{{RoundBoxBottom}}
|Advocating change
Envisioning change

Encouraging innovation

Facilitating collective learning
|The '''primary objective''' is to increase innovation, collective
learning, and adaption to the external environment. 

''Related significantly to managerial effectiveness.'' 
|-
|{{RoundBoxTop|theme=5}}'''External'''  {{RoundBoxBottom}}
|Networking
External monitoring

Representing
|The '''primary objective''' is to acquire the necessary information and resources, and to promote and defend the interests of the team and organisation. 
|}

=== Skills ===
What are leadership skills? According to Northhouse (2021) leadership skills are a set of qualities, capabilities and knowledge, that can be learned and developed. Research findings on capabilities by Mumford and colleagues (2000) identifies five components that are related to effective leadership, they are individual attributes, competencies, leadership outcomes, career experiences, and environmental influences (see Table 2). Mumford et al. (2000) suggest that leadership is not solely dependent on innate traits or personality characteristics but can be developed through the acquisition and refinement of specific skills and knowledge. This perspective aligns with the idea that leadership is a dynamic and learnable process, and individuals can become more effective leaders by focusing on skill development and continuous learning.

{| class=&quot;wikitable&quot;
|+Table 2: Five components related to effective leadership (Northhouse, 2021; Mumford et al., 2000)
|{{RoundBoxTop|theme=2}}'''Individual attributes'''{{RoundBoxBottom}}
|
* General cognitive ability
* Motivation
* Personality
* Crystallised cognitive ability
|These attributes support people 
when applying their leadership competencies.  
|-
|{{RoundBoxTop|theme=3}}'''Competencies'''{{RoundBoxBottom}}
|
* Problem-solving skills
* Social judgement skills
* Knowledge
|These three competencies are important 
elements of effective leadership.
|-
|{{RoundBoxTop|theme=4}}'''Leadership outcomes'''{{RoundBoxBottom}}
|
* Effective problem-solving
* Performance
|These two outcomes are strongly influenced by a leader's competencies. When a leader achieves these outcomes they increase their chances of effective leadership.
|-
|{{RoundBoxTop|theme=5}}'''Career experiences'''{{RoundBoxBottom}}
|
* Challenging projects
* Mentoring and coaching
* Training and development
* Solving complex problems
|A leader's career experience may influence
their problem-solving skills.
|-
|{{RoundBoxTop|theme=6}}'''Environmental influences'''{{RoundBoxBottom}} 
|
* Technology 
* Infrastructure
* Expertise of a team (employees)
* Communication
* Pandemics
|These elements lie outside a leader's capabilities, traits, and experiences.
|}

=== Traits ===
What are leadership traits? Traits refer to people’s general characteristics, which are relatively stable and consistent over time, they include, motives, attitudes, and patterns of behaviour (Kirkpatrick &amp; Locke, 1991). Research conducted by Kirkpatrick and Locke (1991) reveals a set of core characteristics that distinguish successful leaders from others. While these core traits alone do not exclusively determine a person's potential as a leader or a successful one, they are, however, considered prerequisites for people with leadership potential. Kirkpatrick and Locke's (1991) research identified six core traits (see Table 3) in effective leadership. Other traits such as charisma, creativity, and flexibility show inconclusive evidence regarding their importance in effective leadership (Kirkpatrick &amp; Locke, 1991).

{| class=&quot;wikitable&quot;
|+Table 3: Leadership traits (Black &amp; Bright, 2019; Kirkpatrick &amp; Locke, 1991).
!Traits
!Description
|-
|'''''Drive'''''
|The motivation and determination that a leader possesses to achieve their goals and lead their team effectively.
|-
|'''''Leadership motivation''''' 
|An intrinsic desire to take on a leadership role and the responsibilities.
|-
|'''''Integrity'''''
|A commitment to trust, transparency, consistency and the pursuit of high standards of professionalism.
|-
|'''''Self-confidence'''''
|Having a belief in one's own abilities, skills, and judgments. 
|-
|'''''Cognitive ability'''''
|The capability of exercising good judgment, having strong problem-solving and analytical reasoning, possessing the capacity to think strategically and multidimensionally.
|-
|'''''Knowledge of the business'''''
|Having a high degree of understanding of the organisations{{gr}} operating environment and technical oversight.
|}
{{robelbox|theme=13|title=In review}}  &lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
{| class=&quot;wikitable&quot; 
|-
|
{| 
|-
|'''Checking understanding'''                
| 1. How are leadership behaviors classified in the hierarchical taxonomy by Yukl (1999)?
|-
| 2. According to Mumford and colleagues (2000), what are the five components related to effective leadership?
|-
| 3. What are leadership traits, and how do they differ from leadership behaviors and skills?
|}
'''Critical thinking'''
{|
| How do the identified leadership behaviors, skills, and traits interrelate and contribute to effective leadership, and can one compensate for deficiencies in the other?
|}
|}
&lt;/div&gt;
{{Robelbox/close}}

== What are strategies for developing effective leaders? ==
Many organisations are actively enhancing their support for leaders by offering improved development programs, equipping them with new tools and technologies to enhance productivity, and implementing more comprehensive well-being initiatives. Some of these initiatives and strategies for developing effective leaders include:
* Individual [https://www.mindtools.com/a1plnmq/what-is-coaching coaching] aimed at addressing business challenges and fostering professional growth.
* [https://www.mindtools.com/a78j7m1/360-degree-feedback 360-degree assessments] to assist with identifying areas of growth and understanding how one's leadership qualities are perceived by others.
* [https://www.exed.hbs.edu/leadership-development Executive leadership programs] and advanced degree options tailored to develop business expertise and critical industry-specific competencies.
* Using [https://www.apsc.gov.au/initiatives-and-programs/workforce-information/workforce-data/aps-employment-database-interactive-interface-apsedii workforce data and analytics] to provide insights to leaders on their effectiveness as well as workforce management strategies.

==Conclusion==
What does it take to be an effective leader? Essentially, effective leadership is a complex and multifaceted concept that involves a combination of behaviours, skills, and traits. Through the case study of Sarah, a Senior Executive Manager, and Jacinda Ardern, former Prime Minister of New Zealand we have explored the key components of ineffective and effective leadership respectively, and their application in real-world scenarios.

Effective leadership encompasses behaviours such as clarifying goals, supporting team members, envisioning change, and fostering relationships within the team. Relation-oriented behaviours can significantly contribute to job satisfaction and productivity. These behaviours can be developed through coaching and continuous learning and development.

Individual attributes, competencies, leadership outcomes, career experiences, and environmental influences all play a role in shaping effective leaders.

Traits such as drive, leadership motivation, integrity, self-confidence, cognitive ability, and knowledge of the business are considered essential for effective leadership. While other traits like charisma and creativity are of less importance{{gr}}.

In the case of Sarah, addressing the current challenges and negative survey results would require her to focus on developing her emotional intelligence. This would involve recognising and understanding her team members' emotions, managing stress, providing feedback, and fostering open communication. By doing so, she can create a more supportive and inclusive work environment, ultimately improving her leadership effectiveness.

==See also==
* [[Motivation and emotion/Book/2019/Organisational change motivation#Self-determination theory|Organisational change motivation]] (Book chapter, 2019)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
* [[wikipedia:Organizational_behavior|Organisational behaviour]] (Wikipedia)
* [[wikipedia:Human_resource_management|Human resource management]] (Wikipedia)
* [[Motivation and emotion/Book/2011/Work motivation and work satisfaction|Work motivation and work satisfaction]] (Book chapter, 2011)
* [[Motivation and emotion/Book/2021/Sustainable leadership|Sustainable leadership]] (Book chapter, 2021)

==References==
{{Hanging indent|1=


Black, J.S., &amp; Bright, D.S. (2019). Organizational Behavior. Houston, Texas. OpenStax. https://openstax.org/books/organizational-behavior/pages/1-introduction

Brackett, M. A., Rivers, S. E., &amp; Salovey, P. (2011). Emotional intelligence: Implications for personal, social, academic, and workplace success. ''Social and personality psychology compass'', 5(1), 88–103. https://doi.org/10.1111/j.1751-9004.2010.00334.x 

Côté, S., Lopes, P. N., Salovey, P., &amp; Miners, C. T. (2010). Emotional intelligence and leadership emergence in small groups. ''The Leadership Quarterly'', 21(3), 496–508. https://doi.org/10.1016/j.leaqua.2010.03.012

David, S. (2016). Emotional agility: Get unstuck, embrace change, and thrive in work and life. Penguin.

Goleman, D. (1995). Emotional intelligence. Why it can matter more than IQ. New York. Bantam Books.

Greenbaum, R. L., Mawritz, M. B., &amp; Eissa, G. (2012). Bottom-line mentality as an antecedent of social undermining and the moderating roles of core self-evaluations and conscientiousness. ''Journal of Applied Psychology'', 97(2), 343. https://doi.org/10.1037/a0025217

Greenbaum, R. L., Mawritz, M. B., &amp; Zaman, N. N. (2023). The Construct of Bottom-Line Mentality: Where We’ve Been and Where We’re Going. ''Journal of Management'', 49(6), 2109–2147. https://doi.org/10.1177/01492063231153135

Horner, M. (1997). Leadership theory: past, present and future. ''Team Performance Management: An International Journal'', 3(4), 270–287. https://doi.org/10.1108/13527599710195402

Kirkpatrick, S. A., &amp; Locke, E. A. (1991). Leadership: do traits matter?. ''Academy of management perspectives'', 5(2), 48–60. https://doi.org/10.5465/ame.1991.4274679

Northouse, P. G. (2021). Leadership: Theory and practice. Sage publications.

Mumford, M. D., Zaccaro, S. J., Harding, F. D., Jacobs, T. O., &amp; Fleishman, E. A. (2000). Leadership skills for a changing world: Solving complex social problems. ''The leadership quarterly'', 11(1), 11–35.https://doi.org/10.1016/S1048-9843(99)00041-7

OpenAI. (2023). ChatGPT (August 3 version) [Large language model]. https://chat.openai.com/chat

Palmer, B., Walls, M., Burgess, Z., &amp; Stough, C. (2001). Emotional intelligence and effective leadership. ''Leadership and Organization development journal'', 22(1), 5–10. https://doi.org/10.1108/01437730110380174

Ryan, R. M., &amp; Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. ''American psychologist'', 55(1), 68. https://doi.org/10.1037/0003-066X.55.1.68

Salovey, P., &amp; Mayer, J. D. (1990). Emotional intelligence. ''Imagination, Cognition and Personality'', 9, 185–211. https://center.uoregon.edu/StartingStrong/uploads/STARTINGSTRONG2016/HANDOUTS/KEY_46201/pub153_SaloveyMayerICP1990_OCR.pdf

Salovey, P., &amp; Grewal, D. (2005). The science of emotional intelligence. ''Current directions in psychological science'', 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381.x

Thomas, A., &amp; Gupta, V. (2021). Social capital theory, social exchange theory, social cognitive theory, financial literacy, and the role of knowledge sharing as a moderator in enhancing financial well-being: from bibliometric analysis to a conceptual framework model. ''Frontiers in Psychology'', 12, 664638.  https://doi.org/10.3389/fpsyg.2021.664638

Yukl, G. (1999). An evaluative essay on current conceptions of effective leadership. ''European journal of work and organizational psychology'', 8(1), 33–48. https://doi.org/10.1080/135943299398429

Yukl, G., Mahsud, R., Prussia, G., &amp; Hassan, S. (2019). Effectiveness of broad and specific leadership behaviors. ''Personnel Review'', 48(3), 774–783. https://doi.org/10.1108/PR-03-2018-0100
}}

==External links==
* [https://www.ted.com/talks/roselinde_torres_what_it_takes_to_be_a_great_leader What does it take to be a great leader?] (Roselinde Torres)
* [https://www.bing.com/videos/search?q=Susan+David+Emotional+Agility+TED+Talk&amp;&amp;view=detail&amp;mid=29B9442619D47F49617629B9442619D47F496176&amp;&amp;FORM=VRDGAR&amp;ru=%2Fvideos%2Fsearch%3F%26q%3DSusan%2BDavid%2BEmotional%2BAgility%2BTED%2BTalk%26FORM%3DVDMHRS What is emotional agility] (Susan David, Ph.D.)
* [https://www.bing.com/videos/search?q=brene+brown+daring+greatly+worksheet&amp;&amp;view=detail&amp;mid=EDBE0B86648B17584230EDBE0B86648B17584230&amp;&amp;FORM=VRDGAR&amp;ru=%2Fvideos%2Fsearch%3Fq%3Dbrene%2Bbrown%2Bdaring%2Bgreatly%2Bworksheet%26FORM%3DHDRSC6 Which four skill sets make the best leaders] (Dr Brene Brown)
* American psychologist Susan David describes the importance of [https://www.youtube.com/watch?v=0_6hu6JLH98 emotional agility in leaders].
* [https://hbr.org/2020/11/anyone-can-learn-to-be-a-better-leader Anyone can learn to be a better leader] (Monique Valcour)
*Organisational psychologist, Adam Grant's [https://adamgrant.net/podcasts/work-life/ podcast, Work Life]
*[https://coachcast.com.au/ Coachcast] provides leadership coaching tips
*[https://www.abc.net.au/news/2023-01-21/jacinda-arderns-battle-with-death-threats-and-online-trolls/101873990 Jacinda Ardern's leadership]

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__TOC__
==Overview==
{{RoundBoxTop|theme=5}}
;Scenario

[[File:Hand washing man.jpg|right|thumb|150px|'''Figure 1'''. Washing hands became a more frequent hygiene behaviour during the COVID-19 pandemic.]]

You became increasingly aware of your own hygiene when the COVID-19 pandemic took over our lives in 2020. In the beginning, you were motivated by the fear of infection and desire to protect your loved ones. You washed your hands more frequently, used hand sanitiser after touching surfaces in public, cleaned your phone, and became more aware of how germs are passed on from person to person. As time passed, these hygiene behaviours became less about COVID-19, and more about feeling clean, easing the anxiety of spreading germs and keeping up with the standards of our society as encouraged during the pandemic. Even though the threat of COVID-19 has decreased immensely, you continue to wash your hands when you arrive home, sanitise after using public transport, and avoid sharing personal items when someone is sick. This displays how emotions and societal expectations can initially motivate our behaviour, while repetition, personal values and habit formations can maintain these behaviours over the course of our lives. {{RoundBoxBottom}}

* Societal expectations can influence engagement of hygiene behaviours
* Education regarding the purpose of hygiene behaviours is important
* Hygiene behaviours can develop into habits over time
* Socio-economic factors can impact the maintenance of personal hygiene
* Emotions can impact hygiene motivation

{{RoundBoxTop|theme=5}}
'''Focus questions'''

# What are the core motivators for personal hygiene?
# How did the COVID-19 pandemic influence motivation to maintain personal hygiene?
# How do societal expectations influence one’s motivation to maintain personal hygiene?
# Are theoretical perspectives contributing to our understanding of motivation towards personal hygiene?
# Do emotions motivate people to maintain hygiene behaviours?
{{RoundBoxBottom}}

== Core motivators of personal hygiene ==
Personal hygiene can be motivated by a range of factors. The core motivators for personal hygiene are explored below. 

=== Societal expectations ===

* Norms, peer pressure, and group identity can shape hygiene behaviours (Laursen &amp; Faur, 2022).
* Some hygiene behaviours are engaged with purely due to conformity or fear of judgement from others as opposed to a healthy practice (Pilli et al., 2024).
* Example using a family scenario, where one child is encouraged to shower daily, and the other every second day. Are people more motivated to engage in personal hygiene when encouraged by those close to them?

=== Education and understanding of hygiene practices ===
[[File:Wash-your-hands-poster-english-508.pdf|thumb|297x297px|'''Figure 2.''' Public health messaging to promote proper hand washing hygiene in schools. ]]

* Knowledge of the spread of disease through public health messaging (see Figure 2) or learning can influence motivation and engagement with behaviours (Bloomfield &amp; Ackerley, 2023).

* Understanding the reasons behind hygiene practices and their effectiveness can motivate individuals to engage with them, especially to avoid contracting a disease.
* During COVID-19, which will be discussed later in this chapter, health messaging provided explanations as to why washing your hands for 20 seconds with warm water and soap was a vital hygiene behaviour to avoid contracting the illness (World Health Organisation, 2025).

=== Habit formation ===

* Hygiene practices can become second nature through repetition and reinforcement (Singh et al., 2024).
* Brushing teeth, showering, and changing clothes are all personal hygiene behaviours that are second nature to many individuals.
* What contributes to formation of habits?

=== Emotions ===

* Disgust and fear of illness can be drivers for hygiene behaviour in the same way depression and executive dysfunction can act as barriers (Ghassemi et al., 2023).
* Explore levels of fear, including for self and loved ones.
* Explore depression and executive dysfunction specific barriers to personal hygiene.

=== COVID-19 as a catalyst for personal hygiene ===
COVID-19 was a global pandemic that took over the world in 2020. From lockdowns to masks, hygiene behaviours changed drastically for individuals across the globe (World Health Organisation, 2025).

* Increased hygiene engagement
** Handwashing as a more regular practice, sanitiser stations in public spaces, mask hygiene (see Figure 3), disinfecting and cleaning of surfaces after use (World Health Organisation, 2025).

[[File:A woman wearing a protective face mask.jpg|thumb|'''Figure 3.''' Woman wearing mask on public transport during COVID-19 pandemic.]]

* Risk perception and emotions
** Perceived threat or vulnerability to COVID-19 for self or loved ones increased practicing of hygiene behaviours. Motivational influence of fear and uncertainty (Miller et al., 2020).

* Compliance and conformity
** Government mandates (quarantine, isolation, etc.) and collective responsibility of community increased motivation to engage in behaviours. Conforming to the majority, where most people engaged in COVID-19 protocols and influenced others to do the same (Miller et al., 2020).

* Normalisation post-pandemic
** In 2026, many hygiene practices that increased in the pandemic have not reduced, and are the new 'normal'. Social distancing has become second nature to the extent where people judge those who stand too close to others. Wearing a mask when sick or feeling vulnerable has become a more common practice, and hand sanitiser is available in more public spaces than before the pandemic (Ali et al., 2023).

== Theoretical perspectives ==
What can assist in understanding the motivation of hygiene from a theoretical perspective?

=== Health belief model ===

* People decide whether to engage or perform a health behaviour based on cognitive evaluations, as explored below (Alyafei &amp; Easton-Carr, 2024).
** Perceived susceptibility
** Perceived severity
** Perceived benefits
** Perceived barriers
** Cues to action
** Self-efficacy
*How can this relate to maintenance of personal hygiene? 
*How does this help us to understand why people are motivated to engage with personal hygiene behaviours? 

=== Theory of planned behaviour ===

* Behaviour is driven by intention, intention is shaped by three core components (see Figure 4):
** Attitude toward the behaviour (is hygiene beneficial or not?)
** Subjective norms (conformity in relation to motivation)
** Perceived behavioural control (will it be performed successfully, involves access to resources etc.)
*Provide an example of this model using the scenario of COVID-19 in overview

[[File:Theory of planned behaviour.png|thumb|327x327px|'''Figure 4.''' Theory of planned behaviour diagram.]]

* These components combine into behavioural intention and can help predict if someone will actually engage in hygiene behaviours (Bosnjak et al., 2020).

{{Robelbox|title=Quiz|theme=6}}&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
&lt;quiz display=simple&gt;
{'''Which theory involves three core components?'''
|type=&quot;()&quot;}

+ Theory of planned behaviour
- Health belief model 
&lt;/quiz&gt;
{{Robelbox/close}}

== Cultural and economic context ==
Recognising standards of hygiene cross-culturally is important to be able to understand how and why people keep up with personal hygiene.

=== Cross-cultural norms ===

* Personal hygiene expectations and standards can differ between cultures. 
* Hand washing, oral care, and even spitting can be expected at one rate in one culture and a completely different rate in the next. 
* Understanding the definition or standard of hygiene in certain cultures can assist in the understanding of motivational factors behind the behaviour (Eriksson et al., 2021).

=== Economic constraints ===

* Access to water, sanitation, and hygiene products can influence individual's engagement and maintenance of hygiene behaviours. 
* Individuals in developing countries may be motivated to engage in hygiene behaviours, but are unable to due to a lack of resources such as hot water, clean clothes, or menstruation products (Rossouw &amp; Ross, 2021).

[[File:Washing hands kids.jpg|thumb|'''Figure 5.''' School children utilising an outdoor tap to wash their hands. ]]

* Figure 5 displays an example hygiene practices in countries without adequate infrastructure.

== Conclusion ==
* There are a variety of motivating factors for personal hygiene, with societal expectations and emotions being highly influential.
* COVID-19 allowed individuals to understand the benefits of some hygiene practices, with routines that were popular during the pandemic remaining in 2026.
* Personal hygiene can differ across socio-cultural backgrounds, and core motivators may look different cross-culturally.
* Theoretical perspectives such as the health belief model and theory of planned behaviour can deepen the understanding behind motivation and maintenance of personal hygiene. 

== See also ==
{{ic|Rename links as shown in Tutorial 2}}

* [[Motivation and emotion/Book/2024/Disgust and hygiene]] (Book chapter, 2024)
* [[Motivation and emotion/Book/2018/Conformity and emotion]] (Book chapter, 2018)
* [[wikipedia:Public_health_mitigation_of_COVID-19|Public health mitigation of COVID-19]] (Wikipedia)

==References==
{{Hanging indent|1= 

Ali, A. S., Mekonen, Y., &amp; Tesfahun, T. (2023). Hygiene behavior and COVID-19 pandemic: Opportunities of COVID-19-Imposed changes in hygiene behavior. ''INQUIRY: The Journal of Health Care Organization, Provision, and Financing, 60''(1). https://doi.org/10.1177/00469580231218421

Alyafei, A., &amp; Carr, R. E. (2024). The health belief model of behavior change. StatPearls Publishing. National Library of Medicine. https://www.ncbi.nlm.nih.gov/books/NBK606120/

Bloomfield, S. F., &amp; Ackerley, L. (2023). Developing better understanding of hygiene is key to developing hygiene behaviour change in home and everyday life settings. ''Perspect Public Health, 144''(6), 175791392311637–175791392311637. https://doi.org/10.1177/17579139231163734

Bosnjak, M., Ajzen, I., &amp; Schmidt, P. (2020). The theory of planned behavior: Selected recent advances and applications. Europe’s Journal of Psychology, 16(3), 352–356. NCBI. https://doi.org/https://doi.org/10.5964/ejop.v16i3.3107

Eriksson, K., Dickins, T. E., &amp; Strimling, P. (2021). Hygiene norms across 56 nations are predicted by self-control values and disease threat. Current Research in Ecological and Social Psychology, 2(1), 100013. https://doi.org/10.1016/j.cresp.2021.100013

Ghassemi, E., Astrid Hasund Thorseth, Karine Le Roch, Heath, T., &amp; White, S. (2023). Mapping the association between mental health and people’s perceived and actual ability to practice hygiene-related behaviours in humanitarian and pandemic crises: A scoping review. ''PLOS ONE, 18''(12). https://doi.org/10.1371/journal.pone.0286494

Laursen, B., &amp; Faur, S. (2022). What does it mean to be susceptible to influence? A brief primer on peer conformity and developmental changes that affect it. ''International Journal of Behavioral Development, 46''(3), 222–237. https://doi.org/10.1177/01650254221084103

Miller, G. J., Hartman, T. K., Levita, L., Martinez, A. P., Mason, L., McBride, O., McKay, R., Murphy, J., Shevlin, M., Stocks, T. V. A., Bennett, K. M., &amp; Bentall, R. P. (2020). Capability, opportunity, and motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak in the United Kingdom. British Journal of Health Psychology, 25(4), 856–864. https://doi.org/10.1111/bjhp.12426

Pilli, L., Veldwijk, J., Swait, J. D., Donkers, B., &amp; de Bekker-Grob, E. W. (2024). Sources and processes of social influence on health-related choices: A systematic review based on a social-interdependent choice paradigm. ''Social Science &amp; Medicine, 361''(1). https://doi.org/10.1016/j.socscimed.2024.117360

Rossouw, L., &amp; Ross, H. (2021). Understanding period poverty: Socio-economic inequalities in menstrual hygiene management in eight low- and middle-income countries. International Journal of Environmental Research and Public Health, 18(5), 2571. https://doi.org/10.3390/ijerph18052571

Singh, B., Murphy, A., Maher, C., &amp; Smith, A. E. (2024). Time to form a habit: A systematic review and meta-analysis of health behaviour habit formation and its determinants. ''Healthcare, 12''(23), 2488. https://doi.org/10.3390/healthcare12232488

World Health Organization. (2025). ''Coronavirus disease (COVID-19).''  https://www.who.int/health-topics/coronavirus#tab=tab_1 }}

==External links==
* [https://www.beyondblue.org.au/ Support if you are experiencing anxiety or depression] (Beyond Blue)
* [https://open.spotify.com/show/5HG094bpiecqetNsvz0ELs Tips for good hygiene podcast] (Spotify - Sandra Keyes)
* [https://hygieneforhealth.org.au/importance/ Why is hygiene so important?] (Hygiene for Health)

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      <text bytes="11820" sha1="i3bnvv6p2y2ccfcftmdx5m4vn79xprd" xml:space="preserve">== Helpful peer reviewed APA 7 article for Hygiene Motivation if you wish to add or remove!! ==

The concept of hygiene motivation is examined in the study by Curtis, V., &amp; Cairncross, S. (2003), which emphasises that people's hygiene behaviours are primarily driven by the desire to avoid disgust rather than by health education alone. The authors contend that disgust is a potent motivator that can be used in public health campaigns to promote better hygiene practices, particularly in prevention of disease transmission. The study underscores the significance of comprehending emotional drivers like disgust in designing more successful health interventions.

Curtis, V., &amp; Cairncross, S. (2003). Effect of washing hands with soap on diarrhoea risk in the community: a systematic review. ''The Lancet Infectious Diseases'', ''3''(5), 275–281. &lt;nowiki&gt;https://doi.org/10.1016/S1473-3099(03)00606-6&lt;/nowiki&gt;  [[User:U3214564|U3214564]] ([[User talk:U3214564|discuss]] • [[Special:Contributions/U3214564|contribs]]) 08:19, 15 August 2024 (UTC)

== Feedback and Suggestions ==

Hey Jack, I really like your book chapter so far! I found this article from Miller in 2020 which covers motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak. I think this would be an interesting topic to explore and touch on within your book chapter. This article focusses primarily on the role of human behaviour in the COVID-19 outbreak and the speed of the spread across the world, and how human hygiene motivation was associated with the progression.

Here is the reference to the article: Gibson Miller, J. (2020). Capability, opportunity, and motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak in the United Kingdom. ''British Journal of Health Psychology'', ''25''(4), 856–864. &lt;nowiki&gt;https://doi.org/10.1111/bjhp.12426&lt;/nowiki&gt;

Good luck! :)

- Alyssia

[[User:Alyssia Myers|Alyssia Myers]] ([[User talk:Alyssia Myers|discuss]] • [[Special:Contributions/Alyssia Myers|contribs]]) 11:15, 15 August 2024 (UTC)

Hello Jack, it looks like you are hard at work on your page, I've adding another figure to your page, hopefully it'll help you flesh out your article !![https://commons.wikimedia.org/wiki/File:OCD_handwash.jpg this is the image], you could use it to talk about personal hygiene, such as washing one's hands !!! Good luck [[User:Gabriel Geld|Gabriel Geld]] ([[User talk:Gabriel Geld|discuss]] • [[Special:Contributions/Gabriel Geld|contribs]]) 08:43, 16 August 2024 (UTC)

&lt;!-- Official topic development feedback --&gt;
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&lt;!-- Title --&gt;
# The title and sub-title are correctly worded; casing has been corrected
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# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
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}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:53, 18 August 2024 (UTC)

== social contribution ==

hey, I like the case study you have used. linking it to how hygiene decline directly impacts him emotionally and mentally. You could expand on the motivation factors leading to this decline.

you could link it to psychological theories and discuss motivational theories like Maslow’s hierarchy of needs or self-determination theory in relation to hygiene maintenance to add to the depth of your chapter. 

you could also expand on methods to increase hygiene motivation with actionable steps.

Great work so far and good luck :) [[User:U3236683|U3236683]] ([[User talk:U3236683|discuss]] • [[Special:Contributions/U3236683|contribs]]) 07:00, 3 October 2024 (UTC)

== General tips  ==

Hello, 

Here are some suggestions to enhance your chapter:

First, consider simplifying John’s scenario to make it more relatable. Clearly outline how his hygiene standards declined due to increased stress. This will help readers connect with his situation better. Make sure your figure captions are clear and directly related to the content, like explaining the relevance of washing hands in Figure 1.r. Make sure your figure captions are clear and directly related to the content, like explaining the relevance of washing hands in Figure 1.

[[User:U3219927|U3219927]] ([[User talk:U3219927|discuss]] • [[Special:Contributions/U3219927|contribs]]) 11:35, 6 October 2024 (UTC)

== Helpful references  ==

Hi there, 

I have a suggestion for some references you may like to look at for your topic!

'''Motivation Hygiene Theory:''' Brenner, V. C., Carmack, C. W., &amp; Weinstein, M. G. (1971). An empirical test of the motivation-hygiene theory. ''Journal of Accounting Research'', 359-366.

'''Application to real world &amp; COVID:''' Gibson Miller, J., Hartman, T. K., Levita, L., Martinez, A. P., Mason, L., McBride, O., ... &amp; Bentall, R. P. (2020). Capability, opportunity, and motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak in the United Kingdom. ''British journal of health psychology'', ''25''(4), 856-864. [[User:Sienna33309|Sienna33309]] ([[User talk:Sienna33309|discuss]] • [[Special:Contributions/Sienna33309|contribs]]) 02:53, 20 August 2026 (UTC)

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
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&lt;!-- Headings --&gt;
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&lt;!-- 2-level --&gt;
# Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]
&lt;!-- Alignment with focus questions --&gt;
# Reasonably good alignment between focus questions and heading structure, but aim for closer alignment
# The COVID section could be a subheading
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
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# Use open- rather then close-ended focus questions
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&lt;!-- Key points--&gt;
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# Excellent – key points are well developed for each section
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad), but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title
&lt;!-- Theory and research --&gt;
# Promising balance of theory and research
&lt;!-- Conclusion --&gt;
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# What are the practical, take-home messages? (address the focus questions)
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&lt;!-- Figure --&gt;
# Excellent - Relevant figure(s) presented, captioned, and cited
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&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
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&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
# Only cite sources that you have consulted
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# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## italicisation
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}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:22, 31 August 2026 (UTC)</text>
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  <page>
    <title>African Arthropods/Crabroninae</title>
    <ns>0</ns>
    <id>316633</id>
    <revision>
      <id>2829835</id>
      <parentid>2829394</parentid>
      <timestamp>2026-08-31T06:13:27Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* South African wasps of Tribe Larrini */</comment>
      <origin>2829835</origin>
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      <text bytes="3111" sha1="158ovmwq9p8b64l5sie2iu5c4xfuhio" xml:space="preserve">= Afrotropical Crabronidae =

All [[w:Crabronidae|crabronid]] species known in the [[w:Afrotropical realm|Afrotropics]] are of the subfamily Crabroninae. Several species of the subfamily [[w:Dinetinae|Dinetinae]] are found in Egypt, Tunisia and Algeria ([[w:Palearctic realm|Palearctic]]).&lt;ref name=waspweb&gt;van Noort, S. 2024. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/Crabronidae/Classification/index.htm (accessed on 5 February 2025)&lt;/ref&gt;

==South African wasps of Tribe Crabronini==
&lt;gallery mode=packed heights=200&gt;
Dasyproctus iN 27385880 a.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Dasyproctus|''Dasyproctus'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Larrini==
Larrini have:
*Compound eyes that converge above (sometimes parallel in ''Parapiagetia'' and ''Gastrosericus'');
*Posterior ocelli reduced to scars.
For a detailed discussion of the tribe see [https://archive.org/details/bub_gb_FExMjuRhjpIC/page/225/mode/2up Bohart &amp; Menke (1976:226-285)]
See also: [[African Arthropods/South African wasp species of Subtribe Larrina|South African wasp species of Subtribe Larrina]]
&lt;gallery mode=packed heights=200&gt;
Dicranorhina kohli.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps (''[[w:Dicranorhina|Dicranorhina kohli]]'')
Liris haemorrhoidalis iN 47526569 04.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Liris  |''Liris haemorrhoidalis'']])
Liris on Crassula iN 42678436 01.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Liris  |''Liris haemorrhoidalis'']])
Tachysphex specie edit1.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' sp.]])
Tachysphex iN 250449986 2024 10 09 7305.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']])
Tachytes iN 188902572 1964.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachytes |''Tachytes'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Miscophini==
&lt;gallery mode=packed heights=200&gt;
Miscophus kriechbaumeri 2026-03-05.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Miscophus |''Miscophus kriechbaumeri'']])
Paranysson iN 199673489 1.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Paranysson|''Paranysson'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Oxybelini==
&lt;gallery mode=packed heights=200&gt;
Oxybelus iN 250449990 2024 10 09 - 02.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Oxybelus|''Oxybelus'' sp.]])
Oxybelus iN 250450006 2024-10-09 07.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Oxybelus|''Oxybelus'' sp.]])
&lt;/gallery&gt;


==South African wasps of Tribe Palarini==
&lt;gallery mode=packed heights=200&gt;
Palarus_Bee_Pirate_iN_144133368_2022-12-01_01.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Palarus|''Palarus pentheri'']])
&lt;/gallery&gt;


==South African wasps of Tribe Trypoxylini==
&lt;gallery mode=packed heights=200&gt;
Pison iN 144131685 2022-11-30 03.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Pison|''Pison'' sp.]])
Trypoxylon iN 99063113 a.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Trypoxylon|''Trypoxylon'' sp.]])
&lt;/gallery&gt;

==References==
{{reflist}}

{{BookCat}}</text>
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      <text bytes="3116" sha1="rzbcn0ymiye7n9quz5dsuksavhxy9i1" xml:space="preserve">= Afrotropical Crabronidae =

All [[w:Crabronidae|crabronid]] species known in the [[w:Afrotropical realm|Afrotropics]] are of the subfamily Crabroninae. Several species of the subfamily [[w:Dinetinae|Dinetinae]] are found in Egypt, Tunisia and Algeria ([[w:Palearctic realm|Palearctic]]).&lt;ref name=waspweb&gt;van Noort, S. 2024. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/Crabronidae/Classification/index.htm (accessed on 5 February 2025)&lt;/ref&gt;

==South African wasps of Tribe Crabronini==
&lt;gallery mode=packed heights=200&gt;
Dasyproctus iN 27385880 a.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Dasyproctus|''Dasyproctus'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Larrini==
Larrini have:
*Compound eyes that converge above (sometimes parallel in ''Parapiagetia'' and ''Gastrosericus'');
*Posterior ocelli reduced to scars.
For a detailed discussion of the tribe see [https://archive.org/details/bub_gb_FExMjuRhjpIC/page/225/mode/2up Bohart &amp; Menke (1976:226-285)]&lt;br&gt;

See also: [[African Arthropods/South African wasp species of Subtribe Larrina|South African wasp species of Subtribe Larrina]]
&lt;gallery mode=packed heights=200&gt;
Dicranorhina kohli.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps (''[[w:Dicranorhina|Dicranorhina kohli]]'')
Liris haemorrhoidalis iN 47526569 04.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Liris  |''Liris haemorrhoidalis'']])
Liris on Crassula iN 42678436 01.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Liris  |''Liris haemorrhoidalis'']])
Tachysphex specie edit1.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' sp.]])
Tachysphex iN 250449986 2024 10 09 7305.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']])
Tachytes iN 188902572 1964.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachytes |''Tachytes'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Miscophini==
&lt;gallery mode=packed heights=200&gt;
Miscophus kriechbaumeri 2026-03-05.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Miscophus |''Miscophus kriechbaumeri'']])
Paranysson iN 199673489 1.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Paranysson|''Paranysson'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Oxybelini==
&lt;gallery mode=packed heights=200&gt;
Oxybelus iN 250449990 2024 10 09 - 02.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Oxybelus|''Oxybelus'' sp.]])
Oxybelus iN 250450006 2024-10-09 07.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Oxybelus|''Oxybelus'' sp.]])
&lt;/gallery&gt;


==South African wasps of Tribe Palarini==
&lt;gallery mode=packed heights=200&gt;
Palarus_Bee_Pirate_iN_144133368_2022-12-01_01.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Palarus|''Palarus pentheri'']])
&lt;/gallery&gt;


==South African wasps of Tribe Trypoxylini==
&lt;gallery mode=packed heights=200&gt;
Pison iN 144131685 2022-11-30 03.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Pison|''Pison'' sp.]])
Trypoxylon iN 99063113 a.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Trypoxylon|''Trypoxylon'' sp.]])
&lt;/gallery&gt;

==References==
{{reflist}}

{{BookCat}}</text>
      <sha1>rzbcn0ymiye7n9quz5dsuksavhxy9i1</sha1>
    </revision>
    <revision>
      <id>2829837</id>
      <parentid>2829836</parentid>
      <timestamp>2026-08-31T06:26:19Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* South African wasps of Tribe Larrini */</comment>
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      <model>wikitext</model>
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      <text bytes="3094" sha1="10n7b6gmnyip92ow8hk5rfpbj30yt51" xml:space="preserve">= Afrotropical Crabronidae =

All [[w:Crabronidae|crabronid]] species known in the [[w:Afrotropical realm|Afrotropics]] are of the subfamily Crabroninae. Several species of the subfamily [[w:Dinetinae|Dinetinae]] are found in Egypt, Tunisia and Algeria ([[w:Palearctic realm|Palearctic]]).&lt;ref name=waspweb&gt;van Noort, S. 2024. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/Crabronidae/Classification/index.htm (accessed on 5 February 2025)&lt;/ref&gt;

==South African wasps of Tribe Crabronini==
&lt;gallery mode=packed heights=200&gt;
Dasyproctus iN 27385880 a.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Dasyproctus|''Dasyproctus'' sp.]])
&lt;/gallery&gt;

==Tribe Larrini==
Larrini have:
*Compound eyes that converge above (sometimes parallel in ''Parapiagetia'' and ''Gastrosericus'');
*Posterior ocelli reduced to scars.
For a detailed discussion of the tribe see [https://archive.org/details/bub_gb_FExMjuRhjpIC/page/225/mode/2up Bohart &amp; Menke (1976:226-285)]&lt;br&gt;


See also: [[African Arthropods/South African wasp species of Subtribe Larrina|South African wasp species of Subtribe Larrina]]
&lt;gallery mode=packed heights=200&gt;
Dicranorhina kohli.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps (''[[w:Dicranorhina|Dicranorhina kohli]]'')
Liris haemorrhoidalis iN 47526569 04.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Liris  |''Liris haemorrhoidalis'']])
Liris on Crassula iN 42678436 01.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Liris  |''Liris haemorrhoidalis'']])
Tachysphex specie edit1.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' sp.]])
Tachysphex iN 250449986 2024 10 09 7305.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']])
Tachytes iN 188902572 1964.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachytes |''Tachytes'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Miscophini==
&lt;gallery mode=packed heights=200&gt;
Miscophus kriechbaumeri 2026-03-05.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Miscophus |''Miscophus kriechbaumeri'']])
Paranysson iN 199673489 1.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Paranysson|''Paranysson'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Oxybelini==
&lt;gallery mode=packed heights=200&gt;
Oxybelus iN 250449990 2024 10 09 - 02.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Oxybelus|''Oxybelus'' sp.]])
Oxybelus iN 250450006 2024-10-09 07.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Oxybelus|''Oxybelus'' sp.]])
&lt;/gallery&gt;


==South African wasps of Tribe Palarini==
&lt;gallery mode=packed heights=200&gt;
Palarus_Bee_Pirate_iN_144133368_2022-12-01_01.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Palarus|''Palarus pentheri'']])
&lt;/gallery&gt;


==South African wasps of Tribe Trypoxylini==
&lt;gallery mode=packed heights=200&gt;
Pison iN 144131685 2022-11-30 03.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Pison|''Pison'' sp.]])
Trypoxylon iN 99063113 a.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Trypoxylon|''Trypoxylon'' sp.]])
&lt;/gallery&gt;

==References==
{{reflist}}

{{BookCat}}</text>
      <sha1>10n7b6gmnyip92ow8hk5rfpbj30yt51</sha1>
    </revision>
    <revision>
      <id>2829838</id>
      <parentid>2829837</parentid>
      <timestamp>2026-08-31T06:43:41Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Tribe Larrini */</comment>
      <origin>2829838</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4122" sha1="8vpk8y9e4j1898movi7gjuoq9vrf507" xml:space="preserve">= Afrotropical Crabronidae =

All [[w:Crabronidae|crabronid]] species known in the [[w:Afrotropical realm|Afrotropics]] are of the subfamily Crabroninae. Several species of the subfamily [[w:Dinetinae|Dinetinae]] are found in Egypt, Tunisia and Algeria ([[w:Palearctic realm|Palearctic]]).&lt;ref name=waspweb&gt;van Noort, S. 2024. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/Crabronidae/Classification/index.htm (accessed on 5 February 2025)&lt;/ref&gt;

==South African wasps of Tribe Crabronini==
&lt;gallery mode=packed heights=200&gt;
Dasyproctus iN 27385880 a.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Dasyproctus|''Dasyproctus'' sp.]])
&lt;/gallery&gt;

==Tribe Larrini==
Larrini have:
*Compound eyes that converge above (sometimes parallel in ''Parapiagetia'' and ''Gastrosericus'');
*Posterior ocelli reduced to scars.
For a detailed discussion of the tribe see [https://archive.org/details/bub_gb_FExMjuRhjpIC/page/225/mode/2up Bohart &amp; Menke (1976:226-285)]. Genera include ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/251/mode/2up Gastrosericus]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/281/mode/2up Holotachysphex]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/285/mode/2up Kohliella]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/257/mode/2up Larropsis]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/277/mode/2up Parapiagetia]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/281/mode/2up Prosopigastra]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/267/mode/2up Tachysphex]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/255/mode/2up Tachytella]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/259/mode/2up Tachytes]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/249/mode/2up Dicranorhina]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/233/mode/2up Larra]'', and ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/237/mode/2up Liris]''.&lt;br&gt;


See also: [[African Arthropods/South African wasp species of Subtribe Larrina|South African wasp species of Subtribe Larrina]]
&lt;gallery mode=packed heights=200&gt;
Dicranorhina kohli.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps (''[[w:Dicranorhina|Dicranorhina kohli]]'')
Liris haemorrhoidalis iN 47526569 04.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Liris  |''Liris haemorrhoidalis'']])
Liris on Crassula iN 42678436 01.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Liris  |''Liris haemorrhoidalis'']])
Tachysphex specie edit1.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' sp.]])
Tachysphex iN 250449986 2024 10 09 7305.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']])
Tachytes iN 188902572 1964.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachytes |''Tachytes'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Miscophini==
&lt;gallery mode=packed heights=200&gt;
Miscophus kriechbaumeri 2026-03-05.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Miscophus |''Miscophus kriechbaumeri'']])
Paranysson iN 199673489 1.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Paranysson|''Paranysson'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Oxybelini==
&lt;gallery mode=packed heights=200&gt;
Oxybelus iN 250449990 2024 10 09 - 02.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Oxybelus|''Oxybelus'' sp.]])
Oxybelus iN 250450006 2024-10-09 07.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Oxybelus|''Oxybelus'' sp.]])
&lt;/gallery&gt;


==South African wasps of Tribe Palarini==
&lt;gallery mode=packed heights=200&gt;
Palarus_Bee_Pirate_iN_144133368_2022-12-01_01.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Palarus|''Palarus pentheri'']])
&lt;/gallery&gt;


==South African wasps of Tribe Trypoxylini==
&lt;gallery mode=packed heights=200&gt;
Pison iN 144131685 2022-11-30 03.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Pison|''Pison'' sp.]])
Trypoxylon iN 99063113 a.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Trypoxylon|''Trypoxylon'' sp.]])
&lt;/gallery&gt;

==References==
{{reflist}}

{{BookCat}}</text>
      <sha1>8vpk8y9e4j1898movi7gjuoq9vrf507</sha1>
    </revision>
    <revision>
      <id>2829839</id>
      <parentid>2829838</parentid>
      <timestamp>2026-08-31T06:44:38Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Tribe Larrini */</comment>
      <origin>2829839</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4122" sha1="qoig8ppyaz7i8wee7briely179mcm4n" xml:space="preserve">= Afrotropical Crabronidae =

All [[w:Crabronidae|crabronid]] species known in the [[w:Afrotropical realm|Afrotropics]] are of the subfamily Crabroninae. Several species of the subfamily [[w:Dinetinae|Dinetinae]] are found in Egypt, Tunisia and Algeria ([[w:Palearctic realm|Palearctic]]).&lt;ref name=waspweb&gt;van Noort, S. 2024. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/Crabronidae/Classification/index.htm (accessed on 5 February 2025)&lt;/ref&gt;

==South African wasps of Tribe Crabronini==
&lt;gallery mode=packed heights=200&gt;
Dasyproctus iN 27385880 a.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Dasyproctus|''Dasyproctus'' sp.]])
&lt;/gallery&gt;

==Tribe Larrini==
Larrini have:
*Compound eyes that converge above (sometimes parallel in ''Parapiagetia'' and ''Gastrosericus'');
*Posterior ocelli reduced to scars.
For a detailed discussion of the tribe see [https://archive.org/details/bub_gb_FExMjuRhjpIC/page/225/mode/2up Bohart &amp; Menke (1976:226-286)]. Genera include ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/251/mode/2up Gastrosericus]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/281/mode/2up Holotachysphex]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/285/mode/2up Kohliella]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/257/mode/2up Larropsis]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/277/mode/2up Parapiagetia]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/281/mode/2up Prosopigastra]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/267/mode/2up Tachysphex]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/255/mode/2up Tachytella]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/259/mode/2up Tachytes]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/249/mode/2up Dicranorhina]'', ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/233/mode/2up Larra]'', and ''[https://archive.org/details/bub_gb_FExMjuRhjpIC/page/237/mode/2up Liris]''.&lt;br&gt;


See also: [[African Arthropods/South African wasp species of Subtribe Larrina|South African wasp species of Subtribe Larrina]]
&lt;gallery mode=packed heights=200&gt;
Dicranorhina kohli.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps (''[[w:Dicranorhina|Dicranorhina kohli]]'')
Liris haemorrhoidalis iN 47526569 04.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Liris  |''Liris haemorrhoidalis'']])
Liris on Crassula iN 42678436 01.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Liris  |''Liris haemorrhoidalis'']])
Tachysphex specie edit1.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' sp.]])
Tachysphex iN 250449986 2024 10 09 7305.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']])
Tachytes iN 188902572 1964.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachytes |''Tachytes'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Miscophini==
&lt;gallery mode=packed heights=200&gt;
Miscophus kriechbaumeri 2026-03-05.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Miscophus |''Miscophus kriechbaumeri'']])
Paranysson iN 199673489 1.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Paranysson|''Paranysson'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Oxybelini==
&lt;gallery mode=packed heights=200&gt;
Oxybelus iN 250449990 2024 10 09 - 02.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Oxybelus|''Oxybelus'' sp.]])
Oxybelus iN 250450006 2024-10-09 07.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Oxybelus|''Oxybelus'' sp.]])
&lt;/gallery&gt;


==South African wasps of Tribe Palarini==
&lt;gallery mode=packed heights=200&gt;
Palarus_Bee_Pirate_iN_144133368_2022-12-01_01.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Palarus|''Palarus pentheri'']])
&lt;/gallery&gt;


==South African wasps of Tribe Trypoxylini==
&lt;gallery mode=packed heights=200&gt;
Pison iN 144131685 2022-11-30 03.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Pison|''Pison'' sp.]])
Trypoxylon iN 99063113 a.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Trypoxylon|''Trypoxylon'' sp.]])
&lt;/gallery&gt;

==References==
{{reflist}}

{{BookCat}}</text>
      <sha1>qoig8ppyaz7i8wee7briely179mcm4n</sha1>
    </revision>
  </page>
  <page>
    <title>Universal Bibliography/Music</title>
    <ns>0</ns>
    <id>321521</id>
    <revision>
      <id>2829778</id>
      <parentid>2822874</parentid>
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      <contributor>
        <username>James500</username>
        <id>297601</id>
      </contributor>
      <comment>/* Japanese and Japan */ Add</comment>
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      <format>text/x-wiki</format>
      <text bytes="32076" sha1="ivo9uqzcn0hpgkyatjaev5x22iln3w0" xml:space="preserve">{{Bibliography}}

See [[s:Category:Music]] and [[w:Category:Music books]]

This part of the [[Universal Bibliography]] is a bibliography of music.

Bibliography
*[[w:Bibliography of Music Literature|Bibliography of Music Literature]]

*Green (ed). Foundations in Music Bibliography. 1993. [https://books.google.co.uk/books?id=rADdpZN9UhAC&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Krummel. The Literature of Music Bibliography: An Account of the Writings on the History of Music Printing &amp; Publishing. 2nd Ed: 1992. [https://books.google.com/books?id=3AZsiITI-IEC]
*Bibliography of Music Bibliographies. 1967. [https://books.google.co.uk/books?id=d6YJAQAAMAAJ]
*Bayne. A Guide to Library Research in Music. 2008. [https://books.google.co.uk/books?id=ExGbDqu9gPAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*A Selected Bibliography of Music Librarianship [https://books.google.co.uk/books?id=X5AeOl4O-osC]
*Bradley. American Music Librarianship: A Research and Information Guide. [https://books.google.co.uk/books?id=VabcAAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Music Reference and Research Materials. 3rd Ed: 1974: [https://books.google.com/books?id=5Y1IAAAAMAAJ]
*Agruss. Guide to Reference Books on Music. 1948. [https://books.google.co.uk/books?id=wX06AAAAIAAJ]
*Haggerty. A Guide to Popular Music Reference Books: An Annotated Bibliography. 1995. [https://books.google.co.uk/books?id=2OnEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Coover. A Bibliography of Music Dictionaries. 1952: [https://books.google.co.uk/books?id=NH06AAAAIAAJ]. Music Lexicography. 2nd Ed: 1958. Including a Study of Lacunae in Music Lexicography and a Bibliography of Music Dictionaries. 3rd Ed: 1971: [https://books.google.co.uk/books?id=jKMJAQAAMAAJ].
*A Bibliography of Books on Music and Collections of Music. 1948. [https://books.google.co.uk/books?id=vfvpnwWWlZwC]
*Deakin. Musical Bibliography: A Catalogue of the Musical Works. 1892. [https://books.google.co.uk/books?id=-UgQAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false] (England 15th to 18th century)
*Matthew. The Literature of Music. 1896. [https://books.google.co.uk/books?id=fTQ6AAAAMAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]. Reviews: [https://books.google.co.uk/books?id=bjdVAAAAYAAJ&amp;pg=RA1-PA56#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=dzcZAAAAYAAJ&amp;pg=PA22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=R0gcAQAAMAAJ&amp;pg=PA470#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=qK5OAQAAMAAJ&amp;pg=PA55#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1chZAAAAYAAJ&amp;pg=PA155#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=ezszAQAAMAAJ] [https://books.google.co.uk/books?id=5h61TMyTmOMC] [https://books.google.co.uk/books?id=8k8wAQAAIAAJ] [https://books.google.co.uk/books?id=i8W8LKTuc0AC]. Author: [https://books.google.co.uk/books?id=awIQAAAAYAAJ&amp;pg=PA275#v=onepage&amp;q&amp;f=false].
*Hoek. Analyses of Nineteenth- and Twentieth-Century Music, 1940-2000. 2007. [https://books.google.co.uk/books?id=CRG4AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*RILM Abstracts of Music Literature. [https://books.google.co.uk/books?id=HxjjAAAAMAAJ]
*Elliker. The Periodical Literature of Music: Trends from 1952 to 1987. 1996. [https://books.google.co.uk/books?id=T5ifAAAAMAAJ]
*Forkel. Allgemeine Litteratur der Musik. 1792. [https://books.google.co.uk/books?id=VTRDAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false] Review: [https://books.google.co.uk/books?id=3N8sAAAAYAAJ&amp;pg=PA33#v=onepage&amp;q&amp;f=false]

History and bibliography
*Matthew. A Handbook of Musical History and Bibliography. 1898. [https://books.google.co.uk/books?id=V1g5AAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] Review: [https://books.google.co.uk/books?id=P1lDAQAAMAAJ&amp;pg=PA229#v=onepage&amp;q&amp;f=false]
*Boyden. The History and Literature of Music: 1750 to the Present. 1959. [https://books.google.co.uk/books?id=XcAZAQAAIAAJ]
*Brown. An Introduction to the History and Literature of Music in Western Culture. 2nd Ed: 2011. [https://books.google.co.uk/books?id=aKpGAAAACAAJ]

Chronology, annuals, year books, years
*Eisler. World Chronology of Music History.
*Lowe. A Chronological Cyclopædia of Musicians and Musical Events. 1896.
*Tokyo Ongaku Gakko. Kinsei Hogaku Nempyo. [Chronology of Japanese Music in Recent Ages.] Rokugatsu-Kan. Volume 1. 1912. Volume 2. 1914. Volume 3. 1927. [https://books.google.co.uk/books?id=drMQAQAAMAAJ]
*Cossar. This Day in Music. 2005. 2010.
*Glassman. The Year in Music. Columbia House.
*[[w:Herman Klein|Hermann Klein]]. Musical Notes. Annual Critical Record of Important Musical Events.
*[[w:Joseph Bennett (critic)|Bennett]]. The Musical Year.
*Hinrichsen's Musical Year Book
*The Musical Year Book of the United States
**The Boston Musical Year Book
*Billboard. Overview. 1982: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&amp;pg=PT53#v=onepage&amp;q&amp;f=false].
*Billboard. The Year in Music. 1994: [https://books.google.co.uk/books?id=ZAgEAAAAMBAJ&amp;pg=PA62#v=onepage&amp;q&amp;f=false]. 2003: [https://books.google.co.uk/books?id=bA8EAAAAMBAJ&amp;pg=PA47#v=onepage&amp;q&amp;f=false].
**The Year in Music and Video. 1985: [https://books.google.co.uk/books?id=uyQEAAAAMBAJ&amp;pg=PT50#v=onepage&amp;q&amp;f=false]. 1986: [https://books.google.co.uk/books?id=tiQEAAAAMBAJ&amp;pg=PA49#v=onepage&amp;q&amp;f=false].
*Jackson. 1965: The Most Revolutionary Year in Music.
*Porter. A Musical Season: 1972-1973. 
**Music of Three Seasons: 1974-1977 
**Music of Three More Seasons 1977-1980
**Musical Events: A Chronicle, 1980-1983.
*[https://news.1242.com/article/tag/大人のmusic-calendar 【大人のMusic Calendar】]. Nippon Broadcasting System. [Articles from 2016 are included in [https://news.1242.com/article/author/toritani/page/42 NEWS ONLINE 編集部の記事一覧].]
*[http://music-calendar.jp Music Calendar]

History
*&quot;Recorded Sound: The First Century: 1877-1977&quot;. Billboard. 21 May 1977. pp [https://books.google.co.uk/books?id=XCMEAAAAMBAJ&amp;pg=PT39#v=onepage&amp;q&amp;f=false RS-1] to RS-117.

Encyclopedias

See also [[w:List of encyclopedias by branch of knowledge/Music]] and [[w:Bibliography of encyclopedias#Music and dance]]
*Encyclopedia of Music in the 20th Century [https://books.google.co.uk/books?id=m8W2AgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Moore. Complete Encyclopædia of Music. 1852. [https://books.google.co.uk/books?id=-QBFAQAAMAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Dictionaries
*Apel. &quot;Dictionaries of music&quot;. Harvard Dictionary of Music. 1969. pp [https://books.google.co.uk/books?id=TMdf1SioFk4C&amp;pg=PA232#v=onepage&amp;q&amp;f=false 232] to 234.

United Kingdom:
*Billboard. Spotlight on the United Kingdom. 1978:  [https://books.google.co.uk/books?id=TSQEAAAAMBAJ&amp;pg=PT78#v=onepage&amp;q&amp;f=false]. 1979: [https://books.google.co.uk/books?id=MCUEAAAAMBAJ&amp;pg=PT100#v=onepage&amp;q&amp;f=false].

Australia:
*Billboard. Spotlight on Australia/New Zealand. 1982: [https://books.google.co.uk/books?id=GCQEAAAAMBAJ&amp;pg=PT54#v=onepage&amp;q&amp;f=false]. 1985: [https://books.google.co.uk/books?id=hiQEAAAAMBAJ&amp;pg=PT29#v=onepage&amp;q&amp;f=false]. 1986: [https://books.google.co.uk/books?id=UCQEAAAAMBAJ&amp;pg=PA60#v=onepage&amp;q&amp;f=false].
**Live Talent of Australia: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&amp;pg=PT94#v=onepage&amp;q&amp;f=false]

New Zealand:
*Harvey. A Bibliography of Writings about New Zealand Music Published to the End of 1983. 1985. [https://books.google.co.uk/books?id=B1ROA_sP-xsC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*The Complete New Zealand Music Charts, 1966-2006: Singles, Albums, DVDs, Compilations. 2007. [https://books.google.co.uk/books?id=wyU5AQAAIAAJ]
*Billboard. New Zealand. 2002: [https://books.google.co.uk/books?id=Rg0EAAAAMBAJ&amp;pg=PA37#v=onepage&amp;q&amp;f=false]

Canada:
*Billboard. Spotlight on Canada. 1981: [https://books.google.co.uk/books?id=DSQEAAAAMBAJ&amp;pg=PT50#v=onepage&amp;q&amp;f=false].

Scandanavia:
*Billboard. Spotlight on Scandanavia. 1981: [https://books.google.co.uk/books?id=GCUEAAAAMBAJ&amp;pg=PT86#v=onepage&amp;q&amp;f=false].

France:
*Billboard. Spotlight on France. 1971: [https://books.google.co.uk/books?id=-wgEAAAAMBAJ&amp;pg=PA35#v=onepage&amp;q&amp;f=false]. 1972: [https://books.google.co.uk/books?id=REUEAAAAMBAJ&amp;pg=PA35#v=onepage&amp;q&amp;f=false]. 1982: [https://books.google.co.uk/books?id=AyQEAAAAMBAJ&amp;pg=PT66#v=onepage&amp;q&amp;f=false]. 1986: [https://books.google.co.uk/books?id=ICUEAAAAMBAJ&amp;pg=PA41#v=onepage&amp;q&amp;f=false]

Germany:
*Billboard. Spotlight on West Germany. 1971: [https://books.google.co.uk/books?id=zQgEAAAAMBAJ&amp;pg=PA45#v=onepage&amp;q&amp;f=false]. 1985: [https://books.google.co.uk/books?id=-iMEAAAAMBAJ&amp;pg=PT12#v=onepage&amp;q&amp;f=false].
**Spotlight on West Germany, Austria and Switzerland. 1986: [https://books.google.co.uk/books?id=CSUEAAAAMBAJ&amp;pg=RA1-PA35#v=onepage&amp;q&amp;f=false]

Italy:
*Billboard. Spotlight on Italy. 1981: [https://books.google.co.uk/books?id=8iQEAAAAMBAJ&amp;pg=PT3#v=onepage&amp;q&amp;f=false]. 1985: [https://books.google.co.uk/books?id=3yQEAAAAMBAJ&amp;pg=PT36#v=onepage&amp;q&amp;f=false]. 1986: [https://books.google.co.uk/books?id=2SQEAAAAMBAJ&amp;pg=PA38-IA1#v=onepage&amp;q&amp;f=false]. 1994: [https://books.google.co.uk/books?id=XQgEAAAAMBAJ&amp;pg=PA67#v=onepage&amp;q&amp;f=false].

Spain:
*Billboard. Spotlight on Spain. 1971: [https://books.google.co.uk/books?id=5Q8EAAAAMBAJ&amp;pg=PA49#v=onepage&amp;q&amp;f=false]

Philipines:
*[https://billboardphilippines.com/culture/scenes/lost-history-how-filipino-music-was-documented-in-the-40s-to-2010s/ Lost History: How Filipino Music Was Documented In The ’40s To 2010s]. Billboard Philippines. 18 January 2024.
*[[w:en:Billboard Philippines|Billboard Philippines]]

Brazil:
*Billboard. Spotlight on Brazil. 1996: [https://books.google.co.uk/books?id=NA0EAAAAMBAJ&amp;pg=PA51#v=onepage&amp;q&amp;f=false].

United States
*Krummel. Bibliographical Handbook of American Music. 1987. [https://books.google.co.uk/books?id=G4wcnkvFZl4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Krummel. Resources of American Music History: A Directory of Source Materials from Colonial Times to World War II. 1981. [https://books.google.co.uk/books?id=bJcYAAAAIAAJ]

Soviet
*Aschmann. Current Soviet Music Bibliography. 1976. [https://books.google.co.uk/books?id=2i7jAAAAMAAJ]

Decline of pop music:
*[https://www.smithsonianmag.com/smart-news/science-proves-pop-music-has-actually-gotten-worse-8173368/ Science Proves: Pop Music Has Actually Gotten Worse]. [[w:Smithsonian (magazine)|Smithsonian]]. 27 July 2012.
*[https://faroutmagazine.co.uk/new-study-discovers-pop-music-has-suffered-significant-decline-in-one-area/ New study discovers pop music has suffered “significant decline” in one area]. [[w:Far Out (website)|Far Out]]. 5 July 2024.
*[https://www.globalnews.ca/news/9001083/why-older-music-more-popular-than-new-music/amp/ There is something very, very wrong with today’s music. It just may not be very good.] [[w:Global News|Global News]]. 24 July 2022.
*[https://www.bbc.co.uk/music/articles/fb84bf19-29c9-4ed3-b6b6-953e8a083334 Has pop music lost its fun?]. BBC. 12 January 2018.
*[https://www.spectator.co.uk/article/its-official-modern-music-is-bad/ It’s official: modern music is bad]. The Spectator. 13 February 2024.

Homogeneity of pop music:
*[https://www.theguardian.com/music/2012/jul/27/pop-music-sounds-same-survey-reveals Pop music these days: it all sounds the same, survey reveals]. The Guardian. 27 July 2012.
*[https://www.nbcnews.com/id/wbna48356108 Pop Music All Sounds the Same Nowadays]. NBC News. 27 July 2012.
*[https://www.independent.co.uk/voices/comment/why-does-today-s-pop-music-sound-the-same-because-the-same-people-make-it-8368714.html Why does today's pop music sound the same? Because the same people make it]. The Independent. 29 November 2012.
*[https://www.reuters.com/article/lifestyle/science/pop-music-too-loud-and-all-sounds-the-same-official-idUSBRE86P0R9/ Pop music too loud and all sounds the same: official]. Reuters. 26 July 2012.
*[https://theconversation.com/from-art-form-to-asset-our-study-found-popular-songs-are-becoming-more-generic-266097 From art form to asset: our study found popular songs are becoming more generic]. The Conversation. 3 October 2025.

Conferences:
*International Music Industry Conference. 1971: [https://books.google.co.uk/books?id=tggEAAAAMBAJ&amp;pg=PA29#v=onepage&amp;q&amp;f=false]

Laserdisc/Karaoke/CES
*Billboard. Karaoke. 1992:
[https://books.google.co.uk/books?id=jg8EAAAAMBAJ&amp;pg=PA41-IA1#v=onepage&amp;q&amp;f=false]
**CES and Karaoke. 1994. [https://books.google.co.uk/books?id=UggEAAAAMBAJ&amp;pg=PA77#v=onepage&amp;q&amp;f=false]
**Laserdisc. 1995. [https://books.google.co.uk/books?id=7AsEAAAAMBAJ&amp;pg=PA67#v=onepage&amp;q&amp;f=false]
**Laserdisc/Karaoke. 1996: [https://books.google.co.uk/books?id=iQ8EAAAAMBAJ&amp;pg=PA59#v=onepage&amp;q&amp;f=false]

Classical music
*Billboard spotlights: 1995 [https://books.google.co.uk/books?id=1g0EAAAAMBAJ&amp;pg=PA39#v=onepage&amp;q&amp;f=false] (9 September 1995)
**&quot;Classical Music Recording Market&quot;. Billboard. 12 April 1980. pp C-1 to C-12 and p 32. (A Billboard Spotlight).
**&quot;Classical Music: Discovering New Dimensions&quot;. Billboard. 10 September 1983. pp C-1 to C-18. (A Billboard Spotlight).
*&quot;Classical&quot; section, and &quot;Best Selling Classical LPs&quot; chart, in Billboard

Jazz
*[[w:en:All About Jazz|All About Jazz]]

Oldies
*&quot;Oldies stations find their place in radio market&quot;. Star-News. 13 March 1988. pp 1D &amp; [https://books.google.co.uk/books?id=2OoyAAAAIBAJ&amp;pg=PA16#v=onepage&amp;q&amp;f=false 6D]: &quot;Oldies&quot;.
*Billboard. 15 April 1972. [https://books.google.co.uk/books?id=a0UEAAAAMBAJ&amp;pg=PT7#v=onepage&amp;q&amp;f=false p 47].
*Billboard. 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false p 1].
*Billboard. 4 January 1960, [https://books.google.co.uk/books?id=Ch8EAAAAMBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false p 1]

Nostalgia

See also [[Universal Bibliography/Nostalgia]]
*&quot;A Perspective on the Future of Nostalgia&quot;. Billboard.  4 May 1974. pp [https://books.google.co.uk/books?id=cgkEAAAAMBAJ&amp;pg=PA37#v=onepage&amp;q&amp;f=false N-1] to N-54 and two more pages.
*Carr. Nostalgia, Song and the Quest for Home: Production, Text, Reception. 2025. [https://books.google.co.uk/books?id=xz1jEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Charts
*Carroll, &quot; Did Billboard, Cash Box, and Record World Charts Tell the Same Story? Perception and Reality, 1960-1979&quot;(2022) 9 Rock Music Studies [https://www.tandfonline.com/doi/full/10.1080/19401159.2022.2054107 199]

Magazines

See also [[w:Category:Music magazines]]
*Billboard. Google: [https://books.google.co.uk/books/serial/ISSN:00062510?rview=1&amp;lr=&amp;sa=N&amp;start=2770 1942] onwards

==Japanese and Japan==
*The Ashgate Research Companion to Japanese Music. 2017. [https://books.google.co.uk/books?id=W2JTgQGc99EC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=4tINDgAAQBAJ&amp;pg=PA2#v=onepage&amp;q&amp;f=false]

*Billboard. Spotlight on Japan. 1970: 19 December 1970 [https://books.google.co.uk/books?id=mSkEAAAAMBAJ&amp;pg=PA37#v=onepage&amp;q&amp;f=false]. 1971: 11 December 1971 [https://books.google.co.uk/books?id=Fg8EAAAAMBAJ&amp;pg=PA39#v=onepage&amp;q&amp;f=false]. 1973: 17 February 1973 [https://books.google.co.uk/books?id=QEUEAAAAMBAJ&amp;pg=PT25#v=onepage&amp;q&amp;f=false]. 1977: 30 April 1977 [https://books.google.co.uk/books?id=USMEAAAAMBAJ&amp;pg=PT46#v=onepage&amp;q&amp;f=false]. 1979: [https://books.google.co.uk/books?id=_iQEAAAAMBAJ&amp;pg=PT48#v=onepage&amp;q&amp;f=false]. 1982:[https://books.google.co.uk/books?id=byQEAAAAMBAJ&amp;pg=PT38#v=onepage&amp;q&amp;f=false]. 1985: [https://books.google.co.uk/books?id=1CQEAAAAMBAJ&amp;pg=PT65#v=onepage&amp;q&amp;f=false]. 1986:[https://books.google.co.uk/books?id=-CMEAAAAMBAJ&amp;pg=RA1-PA79#v=onepage&amp;q&amp;f=false]. 1993: 12 June 1993 [https://books.google.co.uk/books?id=9A8EAAAAMBAJ&amp;pg=PA57#v=onepage&amp;q&amp;f=false]. 1995: 5 August 1995 [https://books.google.co.uk/books?id=xwsEAAAAMBAJ&amp;pg=PA52-IA1#v=onepage&amp;q&amp;f=false]. 1996: 31 August 1996 [https://books.google.co.uk/books?id=vwcEAAAAMBAJ&amp;pg=PA66#v=onepage&amp;q&amp;f=false]. 1997: 30 August 1997 [https://books.google.co.uk/books?id=_gkEAAAAMBAJ&amp;pg=PA61#v=onepage&amp;q&amp;f=false]. 1998: 26 September 1998 [https://books.google.co.uk/books?id=GgoEAAAAMBAJ&amp;pg=PA117#v=onepage&amp;q&amp;f=false]. 2000: 9 September 2000 [https://books.google.co.uk/books?id=aREEAAAAMBAJ&amp;pg=PA65#v=onepage&amp;q&amp;f=false]. 2002: 7 September 2002 [https://books.google.co.uk/books?id=-QwEAAAAMBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]. 2003: 5 July 2003 [https://books.google.co.uk/books?id=3w0EAAAAMBAJ&amp;pg=PA45#v=onepage&amp;q&amp;f=false].
**&quot;Japan in 1974: Business Bristles While Shortages Are Met&quot;. Billboard. 23 February 1974. pp J-1 to J-30. (A Billboard Spotlight).
**&quot;Made in Japan: A Dynamic Music Industry&quot;. Billboard. 1 March 1975. pp J-1 to J-23. (A Billboard Spotlight).
**&quot;Japan '76&quot;. Billboard. 17 April 1976. pp 36 to 59. (A Billboard Spotlight).
**&quot;Japanese Music: The Challenge of Recession&quot;. Billboard. 27 May 1978. pp J-1 to J-31. (A Billboard Spotlight).
**&quot;Music in Japan: Industry Views 1981 With Quiet Optimism&quot;. Billboard. 30 May 1981. pp J-1 to J-18.
**&quot;Japan: Where Technology Greets Tradition&quot;. (An International Market Profile). Billboard. 21 May 1983. pp J-1 to J-13. Follows p 34.
**&quot;Billboard Spotlight on Japan: VCRs and CDs Will Be Pacemakers&quot;. Billboard. 26 May 1984. pp J-1 to J-11. Follows p 38.
**&quot;Spotlight on Japan&quot;. Billboard. 6 June 1987. pp J-1 to J-12.
**&quot;Japan '88&quot;. Billboard. 9 July 1988. pp J-1 to J-11. (A Billboard International Spotlight).
**&quot;Japan&quot;. (&quot;Japan '89&quot;/&quot;Spotlight on Japan&quot;). Billboard. 3 June 1989. pp J-1 to J-20. (International Spotlight).
**&quot;Japan&quot;. (&quot;International Spotlight&quot;/&quot;A Billboard Spotlight&quot;). Billboard. 25 May 1991. pp J-1 to J-26. Follows p 50. Called &quot;Japan '91&quot; on front page.
*[[w:The Best Ten|The Best Ten]] (ザ・ベストテン). [Television programme]. [https://www.tbs.co.jp/tbs-ch/special/the_bestten/ Episodes].
*[[w:ja:Music Station|Music Station]].  [Television programme]. Episodes: [https://www.tv-asahi.co.jp/music/contents/m_lineup/0003/index.html episode 1] etc.
*Wade. Music in Japan: Experiencing Music, Expressing Culture. 2005. [https://books.google.co.uk/books?id=XXYIAQAAMAAJ]
*Malm. Japanese Music &amp; Musical Instruments. 1959. [https://books.google.com/books?id=QkTaAAAAMAAJ]
*[[w:Francis Taylor Piggott|Pigott]]. The Music and Musical Instruments of Japan. 1893 [https://books.google.co.uk/books?id=ttKTUwmjzMwC&amp;pg=PR3#v=onepage&amp;q&amp;f=false]. 1909. [https://books.google.co.uk/books?id=MAM5AAAAIAAJ]
*Robert C Provine, Yoshiko Tokumaru and J Lawrence Witzleben. &quot;Japan&quot;. East Asia: China, Japan, and Korea. The Garland Encyclopedia of World Music, vol 7. Routledge. 2002. ISBN 0-8240-6041-5. Part 4. pp 531 to 800. [https://books.google.co.uk/books?id=-J1ADwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Toru Mitsui. &quot;Interactions of Imported and Indigenous Musics in Japan: A Historical Overview of the Music Industry&quot;. Fouli T Papageorgiou (ed). Whose Master's Voice?: The Development of Popular Music in Thirteen Cultures. Chapter 9. [https://books.google.co.uk/books?id=xHrDEAAAQBAJ&amp;pg=PA152#v=onepage&amp;q&amp;f=false p 152].

Bibliography
*Tsuge. Japanese Music: An Annotated Bibliography. 1986. [https://books.google.com/books?id=YCsKAQAAMAAJ]
*[[w:ja:三井徹|Tōru Mitsui]]. Popyurā Ongaku Kankei Tosho Mokuroku: Ryūkōka, Jazu, Rokku, J-poppu no Hyakunen. (Japanese: ポピュラー音楽関係図書目録: 流行歌、ジャズ、ロック、Jポップの百年). Nichigai Associates. 2009. [https://books.google.co.uk/books?id=dSAxAQAAIAAJ]. Catalogues: [https://search.worldcat.org/title/406243182] [https://cir.nii.ac.jp/crid/1970586434933272116]
*[https://ndlsearch.ndl.go.jp/rnavi/avmaterials/post_572 音楽に関する文献を探すには（主題書誌）]. NDL.

Dictionaries
*[[w:ja:下中弥三郎|Shimonaka Yasaburo]] (ed). Ongaku Jiten. Heibonsha.  Review: (1959) 18 Journal of Asian Studies 295 [https://www.cambridge.org/core/journals/journal-of-asian-studies/article/abs/ongaku-jiten-dictionary-of-music-ed-shimonaka-yasaburo-tokyo-heibonsha-195557-12-volumes-900-yen-per-volume/F3067B1CE61B5B2C647091E69CE8C8DD] [https://read.dukeupress.edu/journal-of-asian-studies/article-abstract/18/2/295/322980/Ongaku-jiten-Dictionary-of-Music?redirectedFrom=fulltext]

History
*Eta Harich-Schneider. A History of Japanese Music. 1973. [https://books.google.com/books?id=3AraAAAAMAAJ]
*Koh-ichi Hattori. 123 Years of Japanese Music: The Culture of Japan Through a Look at Its Music. 2004. [https://books.google.com/books?id=znzsAAAAMAAJ]
**Koh-ichi Hattori. 36,000 Days of Japanese Music: The Culture of Japan Through A Look At Its Music. Pacific Vision. Pierce, Southfield, Michigan. 1996. ISBN 0965364208.
*Shinpan Nihon Ryūkōkashi. (Japanese: 新版日本流行歌史). [[w:ja:社会思想社|Shakaishisosha]]. 1994. Review: [https://books.google.co.uk/books?id=XQdIAAAAMAAJ]. Catalogue: [https://ndlsearch.ndl.go.jp/en/books/R100000002-I000002420287]
**新版日本流行歌史: 1960-1994. [https://books.google.com/books?id=_b4pAQAAIAAJ] [https://books.google.co.uk/books?id=nb4pAQAAIAAJ]. 
**新版日本流行歌史: 1938-1959
**1867-1937

*Mehl. Music and the Making of Modern Japan: Joining the Global Concert. 2024. [https://books.google.co.uk/books?id=P3QMEQAAQBAJ&amp;pg=PA2#v=onepage&amp;q&amp;f=false]

Annuals, yearbooks
*Ongaku-Nenkan (Japanese: 音楽年鑑) (English: &quot;Music Year Book&quot;, &quot;Music Yearbook&quot;, &quot;Yearbook of Music&quot;). [[w:ja:音楽之友社|Ongaku No Tomo Sha]]. [https://books.google.co.uk/books?id=BhoKAQAAMAAJ 1960] [https://books.google.co.uk/books?id=-bXPAAAAMAAJ 1968] [https://books.google.co.uk/books?id=yNAJAQAAMAAJ 1992]. Catalogues: [https://ci.nii.ac.jp/ncid/BN0029282X] [https://ci.nii.ac.jp/ncid/BN11185209]

Modern, contemporary, today
*Johnson. Handbook of Japanese Music in the Modern Era. 2024. [https://books.google.co.uk/books?id=KNP7EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Matsue. Focus: Music in Contemporary Japan. 2016. [https://books.google.co.uk/books?id=AQgtCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Music of Japan Today. [https://books.google.co.uk/books?id=YZQYEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular music
*Mitsui (ed). Made in Japan: Studies in Popular Music. 2014. [https://books.google.co.uk/books?id=YWQKBAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Stevens. Japanese Popular Music: Culture, Authenticity and Power. 2008. [https://books.google.co.uk/books?id=OHMkdcL9DAMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mitsui. Popular Music in Japan: Transformation Inspired by the West. 2020. [https://books.google.co.uk/books?id=FpbqDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nagahara. Tokyo Boogie-Woogie: Japan’s Pop Era and Its Discontents. 2017. [https://books.google.co.uk/books?id=iTxYDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Patterson. Music and Words: Producing Popular Songs in Modern Japan, 1887–1952. 2019. [https://books.google.co.uk/books?id=P0FvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*James Stanlaw. &quot;Using English identity markers in Japanese Popular Music&quot;. English in East and South Asia. Chapter 14. [https://books.google.co.uk/books?id=88A1EAAAQBAJ&amp;pg=PT109#v=onepage&amp;q&amp;f=false]
*&quot;Japanese Popular Music in Singapore&quot;. Asian Music. vol 34. No 1: Fall/Winter 2002/2003. p 1. [https://books.google.co.uk/books?id=_D4JAQAAMAAJ]
*Steve McClure. Nipponpop. Tuttle Publishing. 1998. ISBN 9780804821070. ISBN 0804821070. [Sometimes called &quot;Nippon Pop&quot;]. Catalogue: [https://search.worldcat.org/title/Nipponpop/oclc/247384040]
Review: (1998) [https://books.google.co.uk/books?id=f9egmeZ8YywC 245] The Publishers Weekly 2 

From folk to J-pop
*[[w:ja:富澤一誠|Issei Tomizawa]]. Ano subarashii kyoku o mō ichido: fōku kara J-poppu made. (Japanese: あの素晴しい曲をもう一度: フォークからJポップまで). [[w:Shinchosha|Shinchosha]]. 2010. [https://books.google.com/books?id=ju9MAQAAIAAJ]. Catalogue: [https://search.worldcat.org/title/501749494]. Commentary on book: [https://www.ytv.co.jp/michiura/time/2010/01/j2010110.html]. Review of the CD: [https://www.cdjournal.com/i/disc/great-agefree-music-forever-and-great-music-are-o/4109110788].

J-pop
*Bourdaghs. Sayonara Amerika, Sayonara Nippon: A Geopolitical Prehistory of J-pop. 2012. [https://books.google.co.uk/books?id=K_y88JwibrMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*&quot;The Rise of J-Pop in Asia and Its Impact&quot; (2004) Japan Spotlight. vol 23. p 24. [https://books.google.co.uk/books?id=i7C0AAAAIAAJ]
*Terence Lancashire. &quot;J-pop's elusive J: Is Japanese popular music Japanese?&quot; (2008) Perfect Beat. vol 9. No 1. p 38. [https://books.google.co.uk/books?id=5No4AQAAIAAJ]
*Tetsu Misaki. J-poppu no Nihongo: kashiron. (Japanese: Jポップの日本語: 歌詞論). [[w:ja:彩流社|彩流社 (Sairyusha)]]. 2002. [https://books.google.com/books?id=dsMpAQAAIAAJ] [https://search.worldcat.org/ja/title/J-:/oclc/52005194]
*[[w:ja:烏賀陽弘道|Hiromichi Ugaya]]. Jpoppu Towa Nanika: Kyodaikasuru Ongaku Sangyō. (Japanese: Jポップとは何か: 巨大化する音楽産業). 2005. [https://books.google.co.uk/books?id=TLlOAAAAMAAJ] catalogue [https://search.worldcat.org/ja/title/J-:/oclc/676652594] [https://ci.nii.ac.jp/ncid/BA71618018]

Japanese rock
*Takarajima Special Edition: Encyclopedia of Japanese Rock 1955-1990. Nihon rokku daihyakka: Rokabirī kara bando būmu made. (Japanese: 日本ロック大百科 [年表編] ロカビリーからバンド・ブームまで 1955〜1990).  [[w:ja:JICC出版局|JICC Shuppankyoku]]. 1992. ISBN 9784796602907. ISBN 4796602909. Catalogues: [https://ci.nii.ac.jp/ncid/BN07889172] [https://catalogue.nla.gov.au/catalog/2263400].
*Japanese Rock: Standard: 1967-1985. 日本のロック名曲徹底ガイド: 名曲263決定盤846. CDJournal. 2008. ISBN 9784861710469. ISBN 4861710464. [https://www.cdjournal.com/Company/products/mook.php?mno=20081002]. Catalogue: [https://ci.nii.ac.jp/ncid/BA8932668X?l=en].
*Kojima Satoshi (Japanese: 小島智). 検証・80年代日本のロック. アルファベータブックス. 2024. ISBN 9784865981179. ISBN 4865981179. [https://books.google.com/books?id=0gbl0AEACAAJ]. Review: [https://mainichi.jp/articles/20241026/ddm/015/070/005000c].

Jazz
*[[w:ja:スイングジャーナル|Swing Journal]] (1947 to 2010) Commentary: [https://www.allaboutjazz.com/news/swing-journal-long-standing-jazz-magazine-to-be-suspended-in-june/] 

Japanese fusion:
*THE DIG presents 日本のフュージョン. Shinko Music Mook. Released 19 April 2013. Commentary: [https://www.cdjournal.com/news/casiopea/50967]. No II. Released 23 October 2014. Commentary: [https://www.cdjournal.com/news/takanaka-masayoshi/62225]

Classical
*[[w:ja:ぶらあぼ|Bravo]] (Japanese: ぶらあぼ) ebravo.jp
*[[w:ja:音楽芸術 (雑誌)|Ongaku Geijutsu]] (Japanese: 音楽芸術)

Magazines

For Japanese music magazines, see [[w:ja:日本の音楽雑誌]].
*Music Periodicals in Japan — A Comprehensive List (1988) 35 Fontes Artis Musicae 116 [https://www.jstor.org/stable/23507222] [https://books.google.com/books?id=qHYWAAAAIAAJ]
**Kishimoto, &quot;Additional Corrections and Alphabetical Title Index&quot; (1989) 36 Fontes Artis Musicae 38 [https://www.jstor.org/stable/23507313] [https://books.google.co.uk/books?id=7XYWAAAAIAAJ] 
*Special Bibliography: A Bibliography of Japanese Magazines and Music (1959) 3 Ethnomusicology 76 [https://www.jstor.org/stable/924290]
*A Historical Survey of Music Periodicals in Japan: 1881—1920 (1989) 36 Fontes Artis Musicae 44 [https://www.jstor.org/stable/23507314]
*[[w:ja:篠原章|Akira Shinohara]]. 日本ロック雑誌クロニクル. [[w:en:Ohta Publishing|Ohta Publishing]]. 2005. [https://books.google.co.uk/books?id=L8opAQAAIAAJ]

*[[w:Oricon|Oricon]] (オリコン)
**[https://web.archive.org/web/19970412131857/http://www.999.com/Oricon/index.html Oricon Music Site]. Commentary: [https://internet.watch.impress.co.jp/www/article/980309/oms.htm].
*[[w:Billboard Japan|Billboard Japan]] (ビルボード・ジャパン)
**Music Labo (ミュージック・ラボ) (1970 to 1994)
*Music Research (ミュージック・リサーチ) [&quot;Weekly Music Magazine&quot;]. Catalogue: [https://web.archive.org/web/20260319070908/https://ndlsearch.ndl.go.jp/books/R100000002-I000000039804].
*Rolling Stone Japan
*新譜ジャーナル (Shinpu Journal). Catalogue: [https://ndlsearch.ndl.go.jp/books/R100000002-I000000012315]. Began 1968 [https://books.google.co.uk/books?id=L8opAQAAIAAJ], later called シンプジャーナル
**シンプジャーナル

*Myūjikku mansurī [ミュージック・マンスリー]  [https://ci.nii.ac.jp/ncid/AN00396190]

*カセットライフ. (Cassette Life). [[w:ja:シンコーミュージック・エンタテイメント|Shinko Music Entertainment]]
*[[w:ja:CDジャーナル|CDJournal]]
*[[w:ja:Rockin'on Japan|Rockin'on Japan]]. (ロッキング・オン・ジャパン). (1986 onwards)
*[[w:ja:Rooftop|Rooftop]] (1976 onwards)

*[[w:ja:FOOL'S MATE|Fool's Mate]]
*Record Monthly (レコード・マンスリー). From 日本レコード振興株式会社. Catalogues: [https://ci.nii.ac.jp/ncid/AA11433275] [https://ndlsearch.ndl.go.jp/books/R100000002-I000000027924]

Columns in periodicals
*&quot;Japanese Newsnotes&quot;. Billboard. (eg 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&amp;pg=PA13#v=onepage&amp;q&amp;f=false p 13].)

Websites
*[[w:ja:ナタリー (ニュースサイト)|Natalie]] (ナタリー)
*[[w:ja:BARKS|Barks]]
*OKMusic. Commentary: [https://xtech.nikkei.com/it/article/NEWS/20120626/405442/].

Charts

For Japanese music charts, see [[w:ja:日本の音楽チャート]]

Chart books
*Oricon Chart Book (Japanese: オリコンチャート・ブック)
**1987 to 1998 Oricon Chart Book. All Albums. [https://books.google.co.uk/books?id=KvEoNwAACAAJ]
**Album Chart Book Complete Edition 1970〜2005. Catalogue:[https://www.tosyokan.pref.shizuoka.jp/licsxp-opac/WOpacMsgNewListToTifTilDetailAction.do?tilcod=1000610247212]
*澤山博之. ミュージック・ライフ 東京で1番売れていたレコード 1958~1966. Shinko Music Entertainment. 2019. [Charts published in Music Life from 1958 onwards]. Commentary: [https://mikiki.tokyo.jp/articles/-/20952 Mikiki]

Number ones
*Oricon No.1 Hits 500. Clubhouse (Japanese: クラブハウス). 1994. 1998.
**[https://books.google.com/books?id=GlsnNwAACAAJ vol 1 (1968~1985)]. ISBN 9784906496129.
**[https://books.google.com/books?id=icInNwAACAAJ vol 2 (1986~1994)]. ISBN 9784906496136.

Awards

Japan Record Awards
*輝く!日本レコード大賞 公式データブック: 放送60回記念: TBS公認. Shinko Music Entertainment. ISBN 9784401647019. [https://books.google.co.uk/books?id=JcDqvwEACAAJ] [https://ci.nii.ac.jp/ncid/BB2773137X]

Traditional, Hogaku
*Malm. Traditional Japanese Music and Musical Instruments. [https://books.google.co.uk/books?id=Yn3VQbqywCsC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Miyuki Yoshikami. Japan's Musical Tradition: Hogaku from Prehistory to the Present. 2020. [https://books.google.co.uk/books?id=X3XTDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Hughes. Traditional Folk Song in Modern Japan: Sources, Sentiment and Society. 2008. [https://books.google.co.uk/books?id=yfV5DwAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

Koto:
*Tokyo Academy of Music. Collection of Japanese Koto Music. 1888. [https://books.google.co.uk/books?id=RncQAAAAYAAJ&amp;pg=PP13#v=onepage&amp;q&amp;f=false][https://babel.hathitrust.org/cgi/pt?id=hvd.32044040839565&amp;seq=1]

Exam guides:

For the 音楽CD検定 exam on music CDs:
*音楽CD検定公式ガイドブック. 2007. [[w:ja:音楽出版社 (企業)|Ongaku Shuppansha Co Ltd]] (音楽出版社). [https://books.google.co.uk/books?id=sbjdeDJMkQcC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=AoFgIowII48C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. Commentary: [https://www.cdjournal.com/i/news/-/15303] [https://www.oricon.co.jp/news/46065/full/] [https://allabout.co.jp/gm/gc/57723/] [https://www.oricon.co.jp/news/45388/full/].

Film
*Tak Shindo. &quot;Japanese Film Music&quot;. Film Journal. No 11: October 1958. p 21. [https://books.google.co.uk/books?id=YF9AIcnhKTIC]

Children's music
*Elizabeth May. The Influence of the Meiji Period on Japanese Children's Music. University of California Press. 1963. [https://books.google.co.uk/books?id=54cHAQAAMAAJ]
**Japanese Children's Music Before and After Contact with the West. University of California at Los Angeles. 1958. (doctoral dissertation).

DJs
*Masahiro Yasuda, &quot;How Japanese DJs cut across Market Boundaries&quot; (1999) [https://books.google.co.uk/books?id=H5QJAQAAMAAJ 4] Perfect Beat 45

[[Category:Music resources]]</text>
      <sha1>ivo9uqzcn0hpgkyatjaev5x22iln3w0</sha1>
    </revision>
  </page>
  <page>
    <title>Bully Metric Realized Timestamps</title>
    <ns>0</ns>
    <id>322040</id>
    <revision>
      <id>2829718</id>
      <parentid>2823242</parentid>
      <timestamp>2026-08-30T12:49:19Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Leap Seconds (1972 - Present) */</comment>
      <origin>2829718</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15300" sha1="p9rfuyiwbpq20j0qjyt00wzsputsyrz" xml:space="preserve">{| class=table style=&quot;width:100%;&quot;
|-
 | {{Original research}}
 | [https://physwiki.eeyabo.net/index.php/Main_Page &lt;small&gt;Development &lt;br/&gt;Area&lt;/small&gt;]
|}

[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt;

There have been over 700,000 realized Bully timestamps during the more than 65 years of modern atomic time keeping (1958 AD ... present). Given the availability of atomic clocks, it is anticipated that Bully timestamps will continue to be realized with great regularity for the foreseeable future.  Each Bully timestamp should be considered &quot;realized&quot; after it occurs and is measured using precise clocks with an accuracy of &lt;math&gt;{10}^{-10}&lt;/math&gt; or better.

=== Leap Seconds (1972 - Present) ===
The below table (derived from the Wikipedia &quot;[[W:Leap Second|Leap Second]]&quot; article), lists all leap second insertions that have occurred since the introduction of leap seconds in 1972.  For each leap second insertion, the below table lists the preceding Bully timestamp (that had been &quot;realized&quot; immediately prior to the leap second insertion), and the subsequent Bully timestamp (that was &quot;realized&quot; immediately after the leap second insertion).

A few details are worth noting in the table.  The TAI and UTC already differed by 10 seconds at the beginning of 1972 due to rubber seconds ([https://en.m.wikiversity.org/wiki/Bully_Metric_Realized_Timestamps#Rubber_Seconds_(1958_-_1971) see discussion below]), so when Bully Timestamp 8209 ECFB E7FB was realized, the TAI time was 1972-06-30 23:34:45 TAI, whereas UTC time was 1972-06-30 23:34:35 UTC. An additional 27 leap seconds were inserted into UTC during the period between 1972 and 2016, making a total of 37 leap seconds difference, so when Bully Timestamp 8209 ED02 EBC0 was realized, the TAI time was 2017-01-01 00:32:00 TAI, whereas UTC time was 2017-01-01 00:31:23 UTC. Notably, Bully timestamps are always realized during TAI times with a seconds value ending in five or zero.  The Bully timestamp and TAI both measure elapsed time as determined by atomic clocks at sea level on Earth, so these systems will always have this simple relationship.  


{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Announced leap seconds to date
|-
! Year !! 30 Jun !! 31 Dec !! Bully Timestamp !! International Atomic Time (TAI) !! Coordinated Universal Time (UTC)
|-
! 1972
|bgcolor=&quot;lime&quot;| +1 ||bgcolor=&quot;lime&quot;| +1 || 8209 ECFB E7FB &lt;br /&gt; 8209 ECFB E7FC &lt;br /&gt; 8209 ECFB FC4F &lt;br /&gt; 8209 ECFB FC50 || 1972-06-30 23:34:45 TAI &lt;br /&gt; 1972-07-01 00:25:40 TAI &lt;br /&gt; 1972-12-31 23:45:05 TAI &lt;br /&gt; 1973-01-01 00:36:00 TAI || 1972-06-30 23:34:35 UTC &lt;br /&gt; 1972-07-01 00:25:29 UTC &lt;br /&gt; 1972-12-31 23:44:54 UTC &lt;br /&gt; 1973-01-01 00:35:48 UTC
|-
! 1973
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ECFC 24A2 &lt;br /&gt; 8209 ECFC 24A3 || 1973-12-31 23:57:50 TAI &lt;br /&gt; 1974-01-01 00:48:45 TAI || 1973-12-31 23:57:38 UTC &lt;br /&gt; 1974-01-01 00:48:32 UTC
|-
! 1974
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ECFC 4CF4 &lt;br /&gt; 8209 ECFC 4CF5 || 1974-12-31 23:19:40 TAI &lt;br /&gt; 1975-01-01 00:10:35 TAI || 1974-12-31 23:19:27 UTC &lt;br /&gt; 1975-01-01 00:10:21 UTC
|-
! 1975
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ECFC 7547 &lt;br /&gt; 8209 ECFC 7548 || 1975-12-31 23:32:25 TAI &lt;br /&gt; 1976-01-01 00:23:20 TAI || 1975-12-31 23:32:11 UTC &lt;br /&gt; 1976-01-01 00:23:05 UTC
|-
! 1976
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ECFC 9DB6 &lt;br /&gt; 8209 ECFC 9DB7 || 1976-12-31 23:30:50 TAI &lt;br /&gt; 1977-01-01 00:21:45 TAI || 1976-12-31 23:30:35 UTC &lt;br /&gt; 1977-01-01 00:21:29 UTC
|-
! 1977
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ECFC C609 &lt;br /&gt; 8209 ECFC C60A || 1977-12-31 23:43:35 TAI &lt;br /&gt; 1978-01-01 00:34:30 TAI || 1977-12-31 23:43:19 UTC &lt;br /&gt; 1978-01-01 00:34:13 UTC
|-
! 1978
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ECFC EE5C &lt;br /&gt; 8209 ECFC EE5D || 1978-12-31 23:56:20 TAI &lt;br /&gt; 1979-01-01 00:47:15 TAI || 1978-12-31 23:56:03 UTC &lt;br /&gt; 1979-01-01 00:46:57 UTC
|-
! 1979
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ECFD 16AE &lt;br /&gt; 8209 ECFD 16AF || 1979-12-31 23:18:10 TAI &lt;br /&gt; 1980-01-01 00:09:05 TAI || 1979-12-31 23:17:52 UTC &lt;br /&gt; 1980-01-01 00:08:46 UTC
|-
! 1981
|bgcolor=&quot;lime&quot;| +1 || 0 || 8209 ECFD 531C &lt;br /&gt; 8209 ECFD 531D || 1981-06-30 23:19:00 TAI &lt;br /&gt; 1981-07-01 00:09:55 TAI || 1981-06-30 23:18:41 UTC &lt;br /&gt; 1981-07-01 00:09:35 UTC
|-
! 1982
|bgcolor=&quot;lime&quot;| +1 || 0 || 8209 ECFD 7B6F &lt;br /&gt; 8209 ECFD 7B70 || 1982-06-30 23:31:45 TAI &lt;br /&gt; 1982-07-01 00:22:40 TAI || 1982-06-30 23:31:25 UTC &lt;br /&gt; 1982-07-01 00:22:19 UTC
|-
! 1983
|bgcolor=&quot;lime&quot;| +1 || 0 || 8209 ECFD A3C2 &lt;br /&gt; 8209 ECFD A3C3 || 1983-06-30 23:44:30 TAI &lt;br /&gt; 1983-07-01 00:35:25 TAI || 1983-06-30 23:44:09 UTC &lt;br /&gt; 1983-07-01 00:35:03 UTC
|-
! 1985
|bgcolor=&quot;lime&quot;| +1 || 0 || 8209 ECFD F484 &lt;br /&gt; 8209 ECFD F485 || 1985-06-30 23:55:40 TAI &lt;br /&gt; 1985-07-01 00:46:35 TAI || 1985-06-30 23:55:18 UTC &lt;br /&gt; 1985-07-01 00:46:12 UTC
|-
! 1987
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ECFE 597D &lt;br /&gt; 8209 ECFE 597E || 1987-12-31 23:40:35 TAI &lt;br /&gt; 1988-01-01 00:31:30 TAI || 1987-12-31 23:40:12 UTC &lt;br /&gt; 1988-01-01 00:31:06 UTC
|-
! 1989
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ECFE AA3F &lt;br /&gt; 8209 ECFE AA40 || 1989-12-31 23:51:45 TAI &lt;br /&gt; 1990-01-01 00:42:40 TAI || 1989-12-31 23:51:21 UTC &lt;br /&gt; 1990-01-01 00:42:15 UTC
|-
! 1990
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ECFE D291 &lt;br /&gt; 8209 ECFE D292 || 1990-12-31 23:13:35 TAI &lt;br /&gt; 1991-01-01 00:04:30 TAI || 1990-12-31 23:13:10 UTC &lt;br /&gt; 1991-01-01 00:04:04 UTC
|-
! 1992
|bgcolor=&quot;lime&quot;| +1 || 0 || 8209 ECFF 0EFF &lt;br /&gt; 8209 ECFF 0F00 || 1992-06-30 23:14:25 TAI &lt;br /&gt; 1992-07-01 00:05:20 TAI || 1992-06-30 23:13:59 UTC &lt;br /&gt; 1992-07-01 00:04:53 UTC
|-
! 1993
|bgcolor=&quot;lime&quot;| +1 || 0 || 8209 ECFF 3752 &lt;br /&gt; 8209 ECFF 3753 || 1993-06-30 23:27:10 TAI &lt;br /&gt; 1993-07-01 00:18:05 TAI || 1993-06-30 23:26:43 UTC &lt;br /&gt; 1993-07-01 00:17:37 UTC
|-
! 1994
|bgcolor=&quot;lime&quot;| +1 || 0 || 8209 ECFF 5FA5 &lt;br /&gt; 8209 ECFF 5FA6 || 1994-06-30 23:39:55 TAI &lt;br /&gt; 1994-07-01 00:30:50 TAI || 1994-06-30 23:39:27 UTC &lt;br /&gt; 1994-07-01 00:30:21 UTC
|-
! 1995
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ECFF 9C4B &lt;br /&gt; 8209 ECFF 9C4C || 1995-12-31 23:12:05 TAI &lt;br /&gt; 1996-01-01 00:03:00 TAI || 1995-12-31 23:11:36 UTC &lt;br /&gt; 1996-01-01 00:02:30 UTC
|-
! 1997
|bgcolor=&quot;lime&quot;| +1 || 0 || 8209 ECFF D8B9 &lt;br /&gt; 8209 ECFF D8BA || 1997-06-30 23:12:55 TAI &lt;br /&gt; 1997-07-01 00:03:50 TAI || 1997-06-30 23:12:25 UTC &lt;br /&gt; 1997-07-01 00:03:19 UTC
|-
! 1998
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ED00 1560 &lt;br /&gt; 8209 ED00 1561 || 1998-12-31 23:36:00 TAI &lt;br /&gt; 1999-01-01 00:26:55 TAI || 1998-12-31 23:35:29 UTC &lt;br /&gt; 1999-01-01 00:26:23 UTC
|-
! 2005
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ED01 2FDC &lt;br /&gt; 8209 ED01 2FDD || 2005-12-31 23:45:40 TAI &lt;br /&gt; 2006-01-01 00:36:35 TAI || 2005-12-31 23:45:08 UTC &lt;br /&gt; 2006-01-01 00:36:02 UTC
|-
! 2008
| 0 ||bgcolor=&quot;lime&quot;| +1 || 8209 ED01 A8F0 &lt;br /&gt; 8209 ED01 A8F1 || 2008-12-31 23:18:40 TAI &lt;br /&gt; 2009-01-01 00:09:35 TAI || 2008-12-31 23:18:07 UTC &lt;br /&gt; 2009-01-01 00:09:01 UTC
|-
! 2012
|bgcolor=&quot;lime&quot;| +1 || 0 || 8209 ED02 3604 &lt;br /&gt; 8209 ED02 3605 || 2012-06-30 23:45:00 TAI &lt;br /&gt; 2012-07-01 00:35:55 TAI || 2012-06-30 23:44:26 UTC &lt;br /&gt; 2012-07-01 00:35:20 UTC
|-
! 2015
|bgcolor=&quot;lime&quot;| +1 || 0 || 8209 ED02 AEFC &lt;br /&gt; 8209 ED02 AEFD || 2015-06-30 23:32:20 TAI &lt;br /&gt; 2015-07-01 00:23:15 TAI || 2015-06-30 23:31:45 UTC &lt;br /&gt; 2015-07-01 00:22:39 UTC
|-
! 2016
| 0 ||bgcolor=&quot;lime&quot;| +1  || 8209 ED02 EBBF &lt;br /&gt; 8209 ED02 EBC0 || 2016-12-31 23:41:05 TAI &lt;br /&gt; 2017-01-01 00:32:00 TAI || 2016-12-31 23:40:29 UTC &lt;br /&gt; 2017-01-01 00:31:23 UTC
|}

=== Rubber Seconds (1958 - 1971) ===
[[File:Bully Timestamps in relation to rubber seconds.png|frame|center|text-bottom|Figure 2: Rubber Seconds]]

Prior to 1972, the rate of UTC atomic clocks was offset from a pure atomic time scale by the BIH to remain synchronized with UT2, a practice known as the &quot;rubber second&quot; (see figure 2). The rate of UTC was decided at the start of each year. Alongside this shift in rate, an occasional 0.1&amp;nbsp;s step (0.05&amp;nbsp;s before 1963) was also implemented as needed.

As shown in figure 2, for 1958-1961, the offset rate was −150 parts per 10{{sup|10}} (or 0.47 seconds per year). This stretching of UTC &quot;rubber seconds&quot; meant that fewer of them would occur during a Bully Timestamp. For example, during the 1958-1961 time period, each Bully timestamp was realized after exactly 3055 seconds TAI, which corresponded to 3054.999955264 seconds UTC. For 1962–63 the offset rate was set to −130 parts per 10{{sup|10}} (or 0.41 seconds per year, or 3054.999960285 seconds UTC per Bully timestamp), and then for 1964–65 the offset rate was returned to −150 parts per 10{{sup|10}}.

The UTC rate of −150 parts per 10{{sup|10}} turned out to be notably inadequate during the 1964-1965 time period, and multiple 0.1&amp;nbsp;s steps were needed (see figure 2). Beginning in 1966, the offset rate was set to −300 parts per 10{{sup|10}} (or 0.94 seconds per year, or 3054.99990835 seconds UTC per Bully timestamp), and this continued until the inauguration of Leap Seconds in 1972.

At the beginning of 1958, the TAI and UTC clocks were in sync, with 1958-01-01 00:00:00.000 TAI occurring at the same time as 1958-01-01 00:00:00.000 UTC.  By the end of 1972, the UTC clock had been adjusted (using rubber seconds and time steps) by ten leap seconds, so that 1972-01-01 00:00:10.003 TAI occurred at the same time as 1972-01-01 00:00:00.003 UTC.  The following table illustrates the slow accumulation of leap seconds prior to 1972, resulting in this ten second difference.

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Rubber Seconds and Accumulative (TAI - UTC) Time Delta
|-
! Approximate Bully Timestamp &lt;br /&gt; Approximate International Atomic Time (TAI) &lt;br /&gt; Coordinated Universal Time (UTC) !! (ΔTAI - ΔUTC) !! Accumulative &lt;br /&gt; Difference
|-
! 8209 ECF9 9F04 (+2820.0 sec) . . .  8209 ECF9 EFAA (+1290.9 sec) &lt;br /&gt;
1958-01-01 00:00:00.000 TAI . . .  1960-01-01 00:00:00.943 TAI &lt;br /&gt;
1958-01-01 00:00:00.002 UTC . . . 1960-01-01 00:00:00.000 UTC
| 0.943 sec || 0.943 sec
|-
! 8209 ECF9 EFAA (+1290.9 sec) . . .  8209 ECFA 1819 (+1386.4 sec) &lt;br /&gt;
1960-01-01 00:00:00.944 TAI . . .  1961-01-01 00:00:01.418 TAI &lt;br /&gt;
1960-01-01 00:00:00.000 UTC . . . 1961-01-01 00:00:00.000 UTC
| 0.474 sec || 1.418 sec
|-
! 8209 ECFA 1819 (+1386.4 sec) &lt;br /&gt;
1961-01-01 00:00:01.418 TAI . . . 1961-01-01 00:00:01.423 TAI &lt;br /&gt;
1961-01-01 00:00:00.000 UTC
| 0.005 sec || 1.423 sec
|-
! 8209 ECFA 1819 (+1386.4 sec) . . .  8209 ECFA 2F85 (+406.6 sec) &lt;br /&gt;
1961-01-01 00:00:01.423 TAI . . .  1961-08-01 00:00:01.698 TAI &lt;br /&gt;
1961-01-01 00:00:00.000 UTC . . . 1961-08-01 00:00:00.000 UTC
| 0.275 sec || 1.698 sec
|-
! 8209 ECFA 2F85 (+406.6 sec) &lt;br /&gt;
1961-08-01 00:00:01.698 TAI . . . 1961-08-01 00:00:01.648 TAI &lt;br /&gt; 
1961-08-01 00:00:00.000 UTC
| -0.050 sec || 1.648 sec
|-
! 8209 ECFA 2F85 (+406.6 sec) . . .  8209 ECFA 406C (+621.8 sec) &lt;br /&gt;
1961-08-01 00:00:01.648 TAI . . .  1962-01-01 00:00:01.846 TAI &lt;br /&gt;
1961-08-01 00:00:00.000 UTC . . . 1962-01-01 00:00:00.000 UTC
| 0.198 sec || 1.846 sec
|-
! 8209 ECFA 406C (+621.8 sec) . . .  8209 ECFA 8A54 (+1622.5 sec) &lt;br /&gt;
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  <page>
    <title>Motivation and emotion/Book/2026/Breathing exercises and relaxation</title>
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      <comment>/* The Promotion of Relaxation Through Breathing Exercises:The Underlying Mechanisms Behind the Ancient Practice. */ added the break in the title</comment>
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      <text bytes="6016" sha1="2ggdiyoj87x5olvb8ox3isdik2lvon2" xml:space="preserve">= The Promotion of Relaxation Through Breathing Exercises:&lt;br&gt;The Underlying Mechanisms Behind the Ancient Practice. =

== Overview ==
{{RoundBoxTop|theme=3}}It’s extremely likely that in a moment of panic or upset you have been told to take a deep breath. Although it may seem pointless or silly in the moment, performing the simple action of taking in a deep breath does more than you think. Go back to a time you were stressed or anxious, and paused for a second to control your breathing. What did you feel in your body? Did you feel more relaxed? Using breathing exercises to promote relaxation is a practice that has been used for thousands of years (Bhargav et al., 2026). 
{{RoundBoxBottom}}

* Provide an Introduction to the topic
* [[File:Falun Dafa fifth meditation exercise.jpg|thumb|'''Figure 1.''' The practice of meditation which is often associated with controlled breathing and relaxation. |316x316px]]brief overview of what will be covered in the chapter


== What is Relaxation? ==

* Definition of [[w:relaxation|relaxation]] including wiki link 
* Introduction to Benson's relaxation response (Benson et al., 1975). 
* Introduction to Jacobson's progressive relaxation theory
* Introduction to ABC relaxation theory (Smith et al., 2000). 

=== Implications for Emotional Regulation ===

* definition of [[w:emotional_regulation|emotional regulation]] including wiki link
* what being able to self regulate can do for reader 
* breathing exercises in emotional regulation (Fu et al., 2026). 

== Parasympathetic Nervous System ==

* About a paragraph explaining in easy to understand terms including wiki link
* Divisions of the nervous system do different things
* sympathetic: fight or flight
* parasympathetic: rest and digest

=== How is it Stimulated? ===

* What is stimulation
* connection to vagus nerve (Yuan &amp; Silberstein, 2016). 
* common ways of stimulating 

=== What happens when it is active? ===

* psychological and physiological changes
* relation back to relaxation theories. 
* Insert quiz: Case study question and answers

== What are Breathing Exercises? ==

* Short definition of breathing exercises 
* brief overview of different types

=== Diaphragmatic breathing ===

* description
* link to external source with instructions: [https://www.youtube.com/watch?v=9jpchJcKivk How to do Diaphragmatic Breathing Exercises for Beginners]

=== Box breathing ===

* description
* link to external source with instructions [https://www.youtube.com/watch?v=mPXy0974vu0 Box Breathing Tutorial | Reduces Anxiety &amp; Stress]

== How do Breathing Exercises Link to Relaxation? ==

* general info and overview about breathing creating relaxation
* mention of practices like yoga and meditation
* mention how mindfullness is often used and taught in clinical settings

=== Parasympathetic stimulation ===

* aligns with this theory of relaxation
* vagus nerve- describe what it is and what it does

=== Cardiovascular and Respiratory Effects ===

* explain decreased heart rate and link to hormones (cortisol)
* what theory of relaxation it aligns with

=== Emotional Regulation ===

* the further effects breathing exercises can have on emotional regulation]
* going back to practicing mindfullness
* link back to theme

== Research ==

* summary of research
* what the reader will get out of understanding
* importance of knowing previous current and future research

=== History ===

* Ancient practices e.g Buddhist meditation (Omelyanenko, 2014) , examples, photos 
* what people originally thought was happening

=== Current ===

* the most influential and current theories surrounding the topic
* at least 2 theories

=== Future ===

* gaps in research
* suggestions for future research
* limitations and implications

== Conclusion ==

* Summary
* Key takeaways relating back to book chapter theme
* Bring back to first learning activity

== See also ==
[[doi:10.1007/978-3-031-56798-8_4.|Slow Breathing for Anxiety: A Critical Perspective Towards Personalization]] (Book Chapter, 2024)

[[wikipedia:Pranayama|Pranyama]] (Wikipedia)

== References ==
Benson, H., Greenwood, M. M., &amp; Klemchuk, H. (1975). The relaxation response: psychophysiologic aspects and clinical applications. ''Int J Psychiatry Med'', ''6''(1-2), 87-98. https://doi.org/10.2190/376w-e4mt-qm6q-h0um.  
 
Bhargav, H., Mehta, U. M., Kumar, A., Subramanian, S., &amp; Keshavan, M. S. (2026). Breathing and the brain: Pranayama, an ancient self-directed approach to neuromodulation. ''Asian Journal of Psychiatry'', ''115'', 104788. https://doi.org/https://doi.org/10.1016/j.ajp.2025.104788. 

Fu, J., Su, Q., &amp; Fu, T. (2026). Effects of breathing rhythm–based exercise intervention on perceived stress and emotion regulation in cancer patients. ''Psycho-oncologie'', ''20''(2), 5706. https://doi.org/10.18282/po5706.  

Omelyanenko, V. I. (2014). Complex integrated method of dynamic meditation with Buddhists’ breathing in case of neurotic reactions. ''Pedagogics, Psychology, Medical-Biological Problems of Physical Training and Sports'', ''18''(2). https://doi.org/10.6084/m9.figshare.923513.   

Smith, J. C., Wedell, A. B., Kolotylo, C. J., Lewis, J. E., Byers, K. Y., &amp; Segin, C. M. (2000). ABC relaxation theory and the factor structure of relaxation states, recalled relaxation activities, dispositions, and motivations. ''Psychol Rep'', ''86''(3 Pt 2), 1201-1208. https://doi.org/10.2466/pr0.2000.86.3c.1201.  

Yuan, H., &amp; Silberstein, S. D. (2016). Vagus Nerve and Vagus Nerve Stimulation, a Comprehensive Review: Part I. ''Headache'', ''56''(1), 71-78. https://doi.org/10.1111/head.12647.   

== External Links ==
[https://www.youtube.com/watch?v=9jpchJcKivk How to do Diaphragmatic Breathing Exercises for Beginners] (youtube video)

[https://www.youtube.com/watch?v=mPXy0974vu0 Box Breathing Tutorial | Reduces Anxiety &amp; Stress] (youtube video)
[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Relaxation]]
[[Category:Motivation and emotion/Book/Stress]]</text>
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  <page>
    <title>Just sustainability transitions: a living review</title>
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    <revision>
      <id>2829717</id>
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      <timestamp>2026-08-30T12:33:02Z</timestamp>
      <contributor>
        <username>Jeanne Noiraud</username>
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      </contributor>
      <comment>/* Future research */ adding future research ideas from qualitative research methodology publication</comment>
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      <format>text/x-wiki</format>
      <text bytes="113805" sha1="fwulasf4kv3jau73p2c4j4yyx4m4x7a" xml:space="preserve">== Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) ==

== Acknowledgements ==
The present text was originally written on a Wikiversity page, if you are reading it in another format, you can find this page here : [[Just sustainability transitions: a living review|https://en.wikiversity.org/wiki/Just_sustainability_transitions:_a_living_review]]. You are free to add your comments on the page discussion section. 

=== Contributors ===
{| class=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-3-030-89460-3_2|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021|publisher=Springer International Publishing|isbn=978-3-030-89460-3|editor-last=Heffron|editor-first=Raphael J.|location=Cham|pages=9–19|language=en|doi=10.1007/978-3-030-89460-3_2}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.sciencedirect.com/science/article/pii/S0301421518302301|journal=Energy Policy|volume=119|pages=1–7|doi=10.1016/j.enpol.2018.04.014|issn=0301-4215}}&lt;/ref&gt;. 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&lt;ref name=&quot;:15&quot;&gt;{{Cite journal|date=2020-03-01|title=Micro-foundations of the multi-level perspective on socio-technical transitions: Developing a multi-dimensional model of agency through crossovers between social constructivism, evolutionary economics and neo-institutional theory|url=https://www.sciencedirect.com/science/article/abs/pii/S0040162518316111|journal=Technological Forecasting and Social Change|language=en-US|volume=152|pages=119894|doi=10.1016/j.techfore.2019.119894|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2023-08-01|title=A socio-technical transition perspective on positive tipping points in climate change mitigation: Analysing seven interacting feedback loops in offshore wind and electric vehicles acceleration|url=https://www.sciencedirect.com/science/article/pii/S0040162523003244|journal=Technological Forecasting and Social Change|language=en-US|volume=193|pages=122639|doi=10.1016/j.techfore.2023.122639|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Sovacool|first=Benjamin K.|last2=Geels|first2=Frank W.|last3=Andersen|first3=Allan Dahl|last4=Grubb|first4=Michael|last5=Jordan|first5=Andrew J.|last6=Kern|first6=Florian|last7=Kivimaa|first7=Paula|last8=Lockwood|first8=Matthew|last9=Markard|first9=Jochen|date=2025-03-01|title=The acceleration of low-carbon transitions: Insights, concepts, challenges, and new directions for research|url=https://www.sciencedirect.com/science/article/pii/S2214629625000295|journal=Energy Research &amp; Social Science|volume=121|pages=103948|doi=10.1016/j.erss.2025.103948|issn=2214-6296}}&lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Kepes|first=Sven|last2=Bennett|first2=Andrew A.|last3=McDaniel|first3=Michael A.|date=2014-09|title=Evidence-Based Management and the Trustworthiness of Our Cumulative Scientific Knowledge: Implications for Teaching, Research, and Practice|url=https://journals.aom.org/doi/10.5465/amle.2013.0193|journal=Academy of Management Learning &amp; Education|volume=13|issue=3|pages=446–466|doi=10.5465/amle.2013.0193|issn=1537-260X}}&lt;/ref&gt;&lt;ref&gt;Pfeffer, J., &amp; Sutton, R. I. (2006). Evidence-Based Management. Harvard Business Review, 13. &lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Bornmann|first=Lutz|last2=Mutz|first2=Rüdiger|date=2015|title=Growth rates of modern science: A bibliometric analysis based on the number of publications and cited references|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/asi.23329|journal=Journal of the Association for Information Science and Technology|language=en|volume=66|issue=11|pages=2215–2222|doi=10.1002/asi.23329|issn=2330-1643}}&lt;/ref&gt;&lt;ref name=&quot;:23&quot;&gt;{{Cite journal|last=Hanson|first=Mark A.|last2=Barreiro|first2=Pablo Gómez|last3=Crosetto|first3=Paolo|last4=Brockington|first4=Dan|date=2024-11-01|title=The strain on scientific publishing|url=https://doi.org/10.1162/qss_a_00327|journal=Quantitative Science Studies|volume=5|issue=4|pages=823–843|doi=10.1162/qss_a_00327|issn=2641-3337}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=http://blogs.nature.com/news/2014/05/global-scientific-output-doubles-every-nine-years.html|title=Global scientific output doubles every nine years : News blog|website=blogs.nature.com|language=en-US|access-date=2026-06-23}}&lt;/ref&gt;, 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&lt;ref name=&quot;:23&quot; /&gt;. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge&lt;ref&gt;{{Cite journal|last=Hartman|first=Yvonne|last2=Darab|first2=Sandy|date=2012-01-01|title=A Call for Slow Scholarship: A Case Study on the Intensification of Academic Life and Its Implications for Pedagogy|url=https://doi.org/10.1080/10714413.2012.643740|journal=Review of Education, Pedagogy, and Cultural Studies|volume=34|issue=1-2|pages=49–60|doi=10.1080/10714413.2012.643740|issn=1071-4413}}&lt;/ref&gt;. “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&lt;ref&gt;{{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)}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=L.|first=Mobley, David|last2=M.|first2=Zuckerman, Daniel|date=2015-02-03|title=A proposal for regularly updated review/survey articles: &quot;Perpetual Reviews&quot;|url=https://arxiv.org/abs/1502.01329|journal=arXiv.org|language=en|doi=10.48550/arXiv.1502.01329}}&lt;/ref&gt; and for shifting from static text format publications to dynamic knowledge mapping&lt;ref name=&quot;:11&quot;&gt;{{Cite web|url=https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/|title=The death of the literature review and the rise of the dynamic knowledge map - LSE Impact|last=Taster|date=2019-05-14|website=LSE Impact - Understanding impact and practice in academic research|access-date=2026-06-23}}&lt;/ref&gt;. This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : &lt;ref&gt;{{Cite journal|last=Elliott|first=Julian H.|last2=Synnot|first2=Anneliese|last3=Turner|first3=Tari|last4=Simmonds|first4=Mark|last5=Akl|first5=Elie A.|last6=McDonald|first6=Steve|last7=Salanti|first7=Georgia|last8=Meerpohl|first8=Joerg|last9=MacLehose|first9=Harriet|date=2017-11|title=Living systematic review: 1. Introduction—the why, what, when, and how|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435617306364|journal=Journal of Clinical Epidemiology|volume=91|pages=23–30|doi=10.1016/j.jclinepi.2017.08.010|issn=0895-4356}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Uttley|first=Lesley|last2=Quintana|first2=Daniel S.|last3=Montgomery|first3=Paul|last4=Carroll|first4=Christopher|last5=Page|first5=Matthew J.|last6=Falzon|first6=Louise|last7=Sutton|first7=Anthea|last8=Moher|first8=David|date=2023-04|title=The problems with systematic reviews: a living systematic review|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435623000112|journal=Journal of Clinical Epidemiology|volume=156|pages=30–41|doi=10.1016/j.jclinepi.2023.01.011|issn=0895-4356}}&lt;/ref&gt;,&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;). 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.

=== 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&lt;ref&gt;{{Cite journal|last=Haley|first=Usha C. V.|date=2023-09-01|title=Triviality and the Search for Scholarly Impact|url=https://doi.org/10.1177/01708406231175292|journal=Organization Studies|language=EN|volume=44|issue=9|pages=1547–1550|doi=10.1177/01708406231175292|issn=0170-8406}}&lt;/ref&gt;. Wikipedia is a key knowledge dissemination platform widely used by students&lt;ref&gt;{{Cite journal|last=Sunvy|first=Ahmed Shafkat|last2=Reza|first2=Raiyan Bin|date=2023-04-12|title=Students’ Perception of Wikipedia as an Academic Information Source|url=https://ejournal.undiksha.ac.id/index.php/IJERR/article/view/57572|journal=Indonesian Journal Of Educational Research and Review|volume=6|issue=1|pages=134–147|doi=10.23887/ijerr.v6i1.57572|issn=2621-8984}}&lt;/ref&gt; and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature&lt;ref&gt;{{Cite journal|last=Thompson|first=Neil|last2=Hanley|first2=Douglas|date=2017|title=Science Is Shaped by Wikipedia: Evidence from a Randomized Control Trial|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3039505|journal=SSRN Electronic Journal|doi=10.2139/ssrn.3039505|issn=1556-5068}}&lt;/ref&gt; and that some scholars cite directly Wikipedia&lt;ref&gt;{{Cite journal|last=Dooley|first=Patricia L.|date=2010-07-07|title=Wikipedia and the two-faced professoriate|url=https://doi.org/10.1145/1832772.1832803|journal=Proceedings of the 6th International Symposium on Wikis and Open Collaboration|series=WikiSym '10|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=1–2|doi=10.1145/1832772.1832803|isbn=978-1-4503-0056-8}}&lt;/ref&gt;. However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives&lt;ref&gt;{{Cite journal|last=Chen|first=Yan|last2=Farzan|first2=Rosta|last3=Kraut|first3=Robert|last4=YeckehZaare|first4=Iman|last5=Zhang|first5=Ark Fangzhou|date=2024-05|title=Motivating Experts to Contribute to Digital Public Goods: A Personalized Field Experiment on Wikipedia|url=https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4852|journal=Management Science|volume=70|issue=5|pages=3264–3280|doi=10.1287/mnsc.2023.4852|issn=0025-1909}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Kincaid|first=Dustin W.|last2=Beck|first2=Whitney S.|last3=Brandt|first3=Jessica E.|last4=Mars Brisbin|first4=Margaret|last5=Farrell|first5=Kaitlin J.|last6=Hondula|first6=Kelly L.|last7=Larson|first7=Erin I.|last8=Shogren|first8=Arial J.|date=2021|title=Wikipedia can help resolve information inequality in the aquatic sciences|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/lol2.10168|journal=Limnology and Oceanography Letters|language=en|volume=6|issue=1|pages=18–23|doi=10.1002/lol2.10168|issn=2378-2242}}&lt;/ref&gt;, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow&lt;ref name=&quot;:10&quot;&gt;Taraborelli, D., Mietchen, D., Alevizou, P., &amp; Gill, A. (2011, August). Expert participation on Wikipedia: Barriers and opportunities. Wikimania 2011, Haifa, Israel. &lt;nowiki&gt;http://upload.wikimedia.org/wikipedia/commons/4/4f/Expert_Participation_Survey_-_Wikimania_2011.pdf&lt;/nowiki&gt; &lt;/ref&gt;. In addition, expert participation is not immune to the gender gap&lt;ref name=&quot;:10&quot; /&gt;. Because of gender segregation in disciplines&lt;ref&gt;{{Cite journal|last=Ceci|first=Stephen J.|last2=Ginther|first2=Donna K.|last3=Kahn|first3=Shulamit|last4=Williams|first4=Wendy M.|date=2014-12-01|title=Women in Academic Science: A Changing Landscape|url=https://doi.org/10.1177/1529100614541236|journal=Psychological Science in the Public Interest|language=EN|volume=15|issue=3|pages=75–141|doi=10.1177/1529100614541236|issn=1529-1006}}&lt;/ref&gt;, this may be detrimental to the content coverage on “female” topics&lt;ref&gt;{{Cite journal|last=Lam|first=Shyong (Tony) K.|last2=Uduwage|first2=Anuradha|last3=Dong|first3=Zhenhua|last4=Sen|first4=Shilad|last5=Musicant|first5=David R.|last6=Terveen|first6=Loren|last7=Riedl|first7=John|date=2011-10-03|title=WP:clubhouse?: an exploration of Wikipedia's gender imbalance|url=https://dl.acm.org/doi/10.1145/2038558.2038560|language=en|publisher=ACM|pages=1–10|doi=10.1145/2038558.2038560|isbn=978-1-4503-0909-7}}&lt;/ref&gt;, 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 &quot;a fair and equitable process of moving towards a post-carbon society&quot;&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;. 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&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 

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&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:16&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; and fragmented results coming from various disciplines&lt;ref name=&quot;:20&quot;&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;. Then, making literature reviews &quot;living&quot; 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&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, 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&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref name=&quot;:12&quot; /&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but not yet integrated into tested and validated methodologies&lt;ref name=&quot;:13&quot;&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;. Human validation stays notably necessary&lt;ref&gt;{{Cite journal|last=Alshami|first=Ahmad|last2=Elsayed|first2=Moustafa|last3=Ali|first3=Eslam|last4=Eltoukhy|first4=Abdelrahman E. E.|last5=Zayed|first5=Tarek|date=2023-07-09|title=Harnessing the Power of ChatGPT for Automating Systematic Review Process: Methodology, Case Study, Limitations, and Future Directions|url=https://www.mdpi.com/2079-8954/11/7/351|journal=Systems|language=en|volume=11|issue=7|pages=351|doi=10.3390/systems11070351|issn=2079-8954}}&lt;/ref&gt;,&lt;ref name=&quot;:13&quot; /&gt;. 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&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&amp;oldid=1360514388|journal=Wikipedia|language=en}}&lt;/ref&gt;, 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, and a &quot;collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation&quot;.&lt;ref&gt;{{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}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&amp;oldid=1360462340|journal=Wikipedia|language=en}}&lt;/ref&gt; &quot;A [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}&lt;/ref&gt; 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.&quot;&lt;ref&gt;{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}&lt;/ref&gt; Such graphs have a strong potential to conduct knowledge synthesis&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;&lt;ref name=&quot;:18&quot; /&gt;. They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;.  

Wikidata is multilingual... 

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]])&lt;ref&gt;{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&amp;oldid=1358552930|journal=Wikipedia|language=en}}&lt;/ref&gt;. Like all Wikimedia projects, Wikiversity is editable by everyone, have a discussion tab and a history log tab. 

=== Use of Wikidata and Wikiversity for research ===

== 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&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt; (e.g. the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;). 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)&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 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=&quot;wikitable&quot;
|+
!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 &quot;export to Wikidata QuickStatements&quot; 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|&quot;Merge&quot; 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. &lt;ref name=&quot;:5&quot; /&gt;). 

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=&quot;wikitable&quot;
|+
!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}}&lt;ref&gt;{{Cite journal|last=Paré|first=Guy|last2=Trudel|first2=Marie-Claude|last3=Jaana|first3=Mirou|last4=Kitsiou|first4=Spyros|date=2015-03|title=Synthesizing information systems knowledge: A typology of literature reviews|url=https://linkinghub.elsevier.com/retrieve/pii/S0378720614001116|journal=Information &amp; Management|language=en|volume=52|issue=2|pages=183–199|doi=10.1016/j.im.2014.08.008}}&lt;/ref&gt;. After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : 
{| class=&quot;wikitable&quot;
|+
!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 &quot;{{Wikidata entity link|Q137901202}}&quot; 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}}, &quot;energy transition&quot; 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 &quot;filter&quot; 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 :&lt;ref&gt;{{Cite journal|last=Nielsen|first=Finn Årup|last2=Lyhne|first2=Ivar|last3=Garigliotti|first3=Dario|last4=Butzbach|first4=Annika|last5=Ravn Boess|first5=Emilia|last6=Hose|first6=Katja|last7=Kørnøv|first7=Lone|date=2023|title=Environmental impact assessment reports in Wikidata and a Wikibase|url=https://repositum.tuwien.at/handle/20.500.12708/193492|language=en|publisher=CEUR-WS.org|volume=3443|pages=1–8|doi=10.34726/5421}}&lt;/ref&gt;)  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)&lt;ref name=&quot;:19&quot;&gt;Cañas, Alberto J., et al. &quot;CmapTools: A knowledge modeling and sharing environment.&quot; (2004): 125-135. https://thomaseskridge.com/assets/pdf/Canas-2004.pdf&lt;/ref&gt;. Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]&lt;ref name=&quot;:19&quot; /&gt;, Altas.Ti &quot;network&quot; feature...). The &quot;box and arrow&quot; 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”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. 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&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[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.&quot;&lt;ref name=&quot;:21&quot;&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; After identifying themes, qualitative researchers are expected to theorize the &quot;arrows&quot; between themes&lt;ref name=&quot;:21&quot; /&gt;R.esearchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;. 

=== 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 &quot;concept&quot; 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&lt;ref name=&quot;:22&quot; /&gt; 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...&lt;ref name=&quot;:22&quot;&gt;{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}&lt;/ref&gt;, and scholars have advised to take insight from philosophy to work on concepts&lt;ref&gt;{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}&lt;/ref&gt;. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt;. 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 &quot;something&quot; 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 &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, 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&amp;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 &quot;does not have effect&quot; seems contrary to what is listed in &quot;has goal&quot;.) We would however argue this is not a problem because &quot;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&quot;&lt;ref&gt;{{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}}&lt;/ref&gt;. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}&lt;/ref&gt;, as long as they are supported by references&lt;ref&gt;{{Cite journal|last=Amaral|first=Gabriel|last2=Piscopo|first2=Alessandro|last3=Kaffee|first3=Lucie-aimée|last4=Rodrigues|first4=Odinaldo|last5=Simperl|first5=Elena|date=2021-12-31|title=Assessing the Quality of Sources in Wikidata Across Languages: A Hybrid Approach|url=https://dl.acm.org/doi/10.1145/3484828|journal=Journal of Data and Information Quality|language=en|volume=13|issue=4|pages=1–35|doi=10.1145/3484828|issn=1936-1955}}&lt;/ref&gt;.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, the discrepancy between the stated goals of this movement and what it actually achieves is precisely what some authors are critiquing&lt;ref name=&quot;:20&quot; /&gt;.  

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 &quot;management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served&quot;&lt;ref&gt;{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research &amp; Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}&lt;/ref&gt;, &quot;choice&quot; 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 &quot;concept&quot; and &quot;political concept&quot;.]]
===== Concision =====
Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were &quot;in the tree of another value&quot;. 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&lt;nowiki/&gt;) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using &quot;instance of&quot; as transversal property and checking &quot;instance of or subclass of&quot;).

*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 &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;&lt;ref&gt;{{Cite journal|last=Vanegas-Cantarero|first=María M.|date=2020-12-01|title=Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries|url=https://www.wikidata.org/wiki/Q137901196|journal=Energy Research &amp; Social Science|language=English|volume=70|doi=10.1016/J.ERSS.2020.101716}}&lt;/ref&gt;. Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for &quot;final energy consumption&quot;. 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&lt;ref&gt;{{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}}&lt;/ref&gt;. 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 &quot;{{Wikidata entity link|Q127270577}}&quot;). 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 &quot;natures&quot; 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)&lt;ref&gt;Fleetwood, S. (2004). An ontology for organisation and management studies. ''Critical Realist Applications in Organisation and Management Studies'', 27–53.&lt;/ref&gt;. 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 &quot;supports multiple coexisting classification&quot; and allow multiple ontological frameworks to coexist.&lt;ref name=&quot;:8&quot; /&gt; 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=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{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}}&lt;/ref&gt; 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.

== 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 statements using a specitic source in order to map the content of a single academic article. First test here: https://w.wiki/PFqH
*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}}).

=== Properties visualisation ===

*Visualise the &quot;tree&quot; 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)

=== Concepts visualisation ===

*Map all statements related to a single item (e.g. [[d:Wikidata:Scholia|Scholia]] request &quot;topic in context&quot;)
*Map the statements that two items have in common to make comparisons.

=== Mapping sources consensus ===

*Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible.

== 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&lt;ref&gt;{{Cite journal|last=Maryl|first=Maciej|last2=Blaszczyńska|first2=Marta|last3=Zalotyńska|first3=Agnieszka|last4=Taylor|first4=Laurence|last5=Avanço|first5=Karla|last6=Balula|first6=Ana|last7=Buchner|first7=Anna|last8=Caliman|first8=Lorena|last9=Clivaz|first9=Claire|date=2021-01|title=Future of Scholarly Communication|url=https://hal.science/hal-03277615}}&lt;/ref&gt;). 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&lt;ref&gt;{{Cite web|url=https://eprints.whiterose.ac.uk/id/eprint/226771/|title=CLOUDSENSE Collaborative Review of Wikipedia Cloud Feedback article|last=Finney|first=D.|last2=Huang|first2=X.|date=2025-05-16|website=eprints.whiterose.ac.uk|language=en|doi=10.48785/100/329|access-date=2026-07-28|last3=Hinnie|first3=L.|last4=Sheppard|first4=N.}}&lt;/ref&gt;.

=== 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 &quot;remixed&quot; 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&lt;ref&gt;{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}&lt;/ref&gt;. Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to &quot;navigate&quot; 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 &quot;navigate&quot; 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&lt;ref name=&quot;:9&quot; /&gt;. 

=== Quality assessment ===
&lt;ref&gt;{{Citation|vauthors=((E. Peireira, A.))|year=2022|title=Usages et transformations d’un commun numérique : les données généalogiques équines|publisher=Paris 10|url=https://theses.fr/s357002|access-date=25 August 2026}}&lt;/ref&gt;  

=== Future research ===
- Can we &quot;reason&quot; 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 &lt;ref name=&quot;:24&quot;&gt;{{cite journal|date=June 2015|title=Novices’ Struggles with Conceptual and Theoretical Framing in Writing Dissertations and Papers for Publication|journal=Publications|publisher=Multidisciplinary Digital Publishing Institute|volume=3|issue=2|pages=104–119|doi=10.3390/publications3020104|issn=2304-6775|vauthors=((Casanave, C. P.)), ((Li, Y.))}}&lt;/ref&gt;. Conceptualization require to answer the question « what are our data examples of? » &lt;ref name=&quot;:24&quot; /&gt;.  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 &lt;ref name=&quot;:24&quot; /&gt;. 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_&amp;_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=&quot;wikitable sortable&quot;
! 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}}</text>
      <sha1>fwulasf4kv3jau73p2c4j4yyx4m4x7a</sha1>
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        <username>Jeanne Noiraud</username>
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      <comment>/* Concepts visualisation */ Adding data visualisation example</comment>
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      <text bytes="114078" sha1="3wxt3eqmxscfdeu6ncsj997b6v60smp" xml:space="preserve">== Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) ==

== Acknowledgements ==
The present text was originally written on a Wikiversity page, if you are reading it in another format, you can find this page here : [[Just sustainability transitions: a living review|https://en.wikiversity.org/wiki/Just_sustainability_transitions:_a_living_review]]. You are free to add your comments on the page discussion section. 

=== Contributors ===
{| class=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-3-030-89460-3_2|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021|publisher=Springer International Publishing|isbn=978-3-030-89460-3|editor-last=Heffron|editor-first=Raphael J.|location=Cham|pages=9–19|language=en|doi=10.1007/978-3-030-89460-3_2}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.sciencedirect.com/science/article/pii/S0301421518302301|journal=Energy Policy|volume=119|pages=1–7|doi=10.1016/j.enpol.2018.04.014|issn=0301-4215}}&lt;/ref&gt;. 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&lt;ref name=&quot;:15&quot;&gt;{{Cite journal|date=2020-03-01|title=Micro-foundations of the multi-level perspective on socio-technical transitions: Developing a multi-dimensional model of agency through crossovers between social constructivism, evolutionary economics and neo-institutional theory|url=https://www.sciencedirect.com/science/article/abs/pii/S0040162518316111|journal=Technological Forecasting and Social Change|language=en-US|volume=152|pages=119894|doi=10.1016/j.techfore.2019.119894|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2023-08-01|title=A socio-technical transition perspective on positive tipping points in climate change mitigation: Analysing seven interacting feedback loops in offshore wind and electric vehicles acceleration|url=https://www.sciencedirect.com/science/article/pii/S0040162523003244|journal=Technological Forecasting and Social Change|language=en-US|volume=193|pages=122639|doi=10.1016/j.techfore.2023.122639|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Sovacool|first=Benjamin K.|last2=Geels|first2=Frank W.|last3=Andersen|first3=Allan Dahl|last4=Grubb|first4=Michael|last5=Jordan|first5=Andrew J.|last6=Kern|first6=Florian|last7=Kivimaa|first7=Paula|last8=Lockwood|first8=Matthew|last9=Markard|first9=Jochen|date=2025-03-01|title=The acceleration of low-carbon transitions: Insights, concepts, challenges, and new directions for research|url=https://www.sciencedirect.com/science/article/pii/S2214629625000295|journal=Energy Research &amp; Social Science|volume=121|pages=103948|doi=10.1016/j.erss.2025.103948|issn=2214-6296}}&lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Kepes|first=Sven|last2=Bennett|first2=Andrew A.|last3=McDaniel|first3=Michael A.|date=2014-09|title=Evidence-Based Management and the Trustworthiness of Our Cumulative Scientific Knowledge: Implications for Teaching, Research, and Practice|url=https://journals.aom.org/doi/10.5465/amle.2013.0193|journal=Academy of Management Learning &amp; Education|volume=13|issue=3|pages=446–466|doi=10.5465/amle.2013.0193|issn=1537-260X}}&lt;/ref&gt;&lt;ref&gt;Pfeffer, J., &amp; Sutton, R. I. (2006). Evidence-Based Management. Harvard Business Review, 13. &lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Bornmann|first=Lutz|last2=Mutz|first2=Rüdiger|date=2015|title=Growth rates of modern science: A bibliometric analysis based on the number of publications and cited references|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/asi.23329|journal=Journal of the Association for Information Science and Technology|language=en|volume=66|issue=11|pages=2215–2222|doi=10.1002/asi.23329|issn=2330-1643}}&lt;/ref&gt;&lt;ref name=&quot;:23&quot;&gt;{{Cite journal|last=Hanson|first=Mark A.|last2=Barreiro|first2=Pablo Gómez|last3=Crosetto|first3=Paolo|last4=Brockington|first4=Dan|date=2024-11-01|title=The strain on scientific publishing|url=https://doi.org/10.1162/qss_a_00327|journal=Quantitative Science Studies|volume=5|issue=4|pages=823–843|doi=10.1162/qss_a_00327|issn=2641-3337}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=http://blogs.nature.com/news/2014/05/global-scientific-output-doubles-every-nine-years.html|title=Global scientific output doubles every nine years : News blog|website=blogs.nature.com|language=en-US|access-date=2026-06-23}}&lt;/ref&gt;, 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&lt;ref name=&quot;:23&quot; /&gt;. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge&lt;ref&gt;{{Cite journal|last=Hartman|first=Yvonne|last2=Darab|first2=Sandy|date=2012-01-01|title=A Call for Slow Scholarship: A Case Study on the Intensification of Academic Life and Its Implications for Pedagogy|url=https://doi.org/10.1080/10714413.2012.643740|journal=Review of Education, Pedagogy, and Cultural Studies|volume=34|issue=1-2|pages=49–60|doi=10.1080/10714413.2012.643740|issn=1071-4413}}&lt;/ref&gt;. “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&lt;ref&gt;{{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)}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=L.|first=Mobley, David|last2=M.|first2=Zuckerman, Daniel|date=2015-02-03|title=A proposal for regularly updated review/survey articles: &quot;Perpetual Reviews&quot;|url=https://arxiv.org/abs/1502.01329|journal=arXiv.org|language=en|doi=10.48550/arXiv.1502.01329}}&lt;/ref&gt; and for shifting from static text format publications to dynamic knowledge mapping&lt;ref name=&quot;:11&quot;&gt;{{Cite web|url=https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/|title=The death of the literature review and the rise of the dynamic knowledge map - LSE Impact|last=Taster|date=2019-05-14|website=LSE Impact - Understanding impact and practice in academic research|access-date=2026-06-23}}&lt;/ref&gt;. This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : &lt;ref&gt;{{Cite journal|last=Elliott|first=Julian H.|last2=Synnot|first2=Anneliese|last3=Turner|first3=Tari|last4=Simmonds|first4=Mark|last5=Akl|first5=Elie A.|last6=McDonald|first6=Steve|last7=Salanti|first7=Georgia|last8=Meerpohl|first8=Joerg|last9=MacLehose|first9=Harriet|date=2017-11|title=Living systematic review: 1. Introduction—the why, what, when, and how|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435617306364|journal=Journal of Clinical Epidemiology|volume=91|pages=23–30|doi=10.1016/j.jclinepi.2017.08.010|issn=0895-4356}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Uttley|first=Lesley|last2=Quintana|first2=Daniel S.|last3=Montgomery|first3=Paul|last4=Carroll|first4=Christopher|last5=Page|first5=Matthew J.|last6=Falzon|first6=Louise|last7=Sutton|first7=Anthea|last8=Moher|first8=David|date=2023-04|title=The problems with systematic reviews: a living systematic review|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435623000112|journal=Journal of Clinical Epidemiology|volume=156|pages=30–41|doi=10.1016/j.jclinepi.2023.01.011|issn=0895-4356}}&lt;/ref&gt;,&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;). 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.

=== 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&lt;ref&gt;{{Cite journal|last=Haley|first=Usha C. V.|date=2023-09-01|title=Triviality and the Search for Scholarly Impact|url=https://doi.org/10.1177/01708406231175292|journal=Organization Studies|language=EN|volume=44|issue=9|pages=1547–1550|doi=10.1177/01708406231175292|issn=0170-8406}}&lt;/ref&gt;. Wikipedia is a key knowledge dissemination platform widely used by students&lt;ref&gt;{{Cite journal|last=Sunvy|first=Ahmed Shafkat|last2=Reza|first2=Raiyan Bin|date=2023-04-12|title=Students’ Perception of Wikipedia as an Academic Information Source|url=https://ejournal.undiksha.ac.id/index.php/IJERR/article/view/57572|journal=Indonesian Journal Of Educational Research and Review|volume=6|issue=1|pages=134–147|doi=10.23887/ijerr.v6i1.57572|issn=2621-8984}}&lt;/ref&gt; and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature&lt;ref&gt;{{Cite journal|last=Thompson|first=Neil|last2=Hanley|first2=Douglas|date=2017|title=Science Is Shaped by Wikipedia: Evidence from a Randomized Control Trial|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3039505|journal=SSRN Electronic Journal|doi=10.2139/ssrn.3039505|issn=1556-5068}}&lt;/ref&gt; and that some scholars cite directly Wikipedia&lt;ref&gt;{{Cite journal|last=Dooley|first=Patricia L.|date=2010-07-07|title=Wikipedia and the two-faced professoriate|url=https://doi.org/10.1145/1832772.1832803|journal=Proceedings of the 6th International Symposium on Wikis and Open Collaboration|series=WikiSym '10|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=1–2|doi=10.1145/1832772.1832803|isbn=978-1-4503-0056-8}}&lt;/ref&gt;. However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives&lt;ref&gt;{{Cite journal|last=Chen|first=Yan|last2=Farzan|first2=Rosta|last3=Kraut|first3=Robert|last4=YeckehZaare|first4=Iman|last5=Zhang|first5=Ark Fangzhou|date=2024-05|title=Motivating Experts to Contribute to Digital Public Goods: A Personalized Field Experiment on Wikipedia|url=https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4852|journal=Management Science|volume=70|issue=5|pages=3264–3280|doi=10.1287/mnsc.2023.4852|issn=0025-1909}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Kincaid|first=Dustin W.|last2=Beck|first2=Whitney S.|last3=Brandt|first3=Jessica E.|last4=Mars Brisbin|first4=Margaret|last5=Farrell|first5=Kaitlin J.|last6=Hondula|first6=Kelly L.|last7=Larson|first7=Erin I.|last8=Shogren|first8=Arial J.|date=2021|title=Wikipedia can help resolve information inequality in the aquatic sciences|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/lol2.10168|journal=Limnology and Oceanography Letters|language=en|volume=6|issue=1|pages=18–23|doi=10.1002/lol2.10168|issn=2378-2242}}&lt;/ref&gt;, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow&lt;ref name=&quot;:10&quot;&gt;Taraborelli, D., Mietchen, D., Alevizou, P., &amp; Gill, A. (2011, August). Expert participation on Wikipedia: Barriers and opportunities. Wikimania 2011, Haifa, Israel. &lt;nowiki&gt;http://upload.wikimedia.org/wikipedia/commons/4/4f/Expert_Participation_Survey_-_Wikimania_2011.pdf&lt;/nowiki&gt; &lt;/ref&gt;. In addition, expert participation is not immune to the gender gap&lt;ref name=&quot;:10&quot; /&gt;. Because of gender segregation in disciplines&lt;ref&gt;{{Cite journal|last=Ceci|first=Stephen J.|last2=Ginther|first2=Donna K.|last3=Kahn|first3=Shulamit|last4=Williams|first4=Wendy M.|date=2014-12-01|title=Women in Academic Science: A Changing Landscape|url=https://doi.org/10.1177/1529100614541236|journal=Psychological Science in the Public Interest|language=EN|volume=15|issue=3|pages=75–141|doi=10.1177/1529100614541236|issn=1529-1006}}&lt;/ref&gt;, this may be detrimental to the content coverage on “female” topics&lt;ref&gt;{{Cite journal|last=Lam|first=Shyong (Tony) K.|last2=Uduwage|first2=Anuradha|last3=Dong|first3=Zhenhua|last4=Sen|first4=Shilad|last5=Musicant|first5=David R.|last6=Terveen|first6=Loren|last7=Riedl|first7=John|date=2011-10-03|title=WP:clubhouse?: an exploration of Wikipedia's gender imbalance|url=https://dl.acm.org/doi/10.1145/2038558.2038560|language=en|publisher=ACM|pages=1–10|doi=10.1145/2038558.2038560|isbn=978-1-4503-0909-7}}&lt;/ref&gt;, 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 &quot;a fair and equitable process of moving towards a post-carbon society&quot;&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;. 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&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 

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&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:16&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; and fragmented results coming from various disciplines&lt;ref name=&quot;:20&quot;&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;. Then, making literature reviews &quot;living&quot; 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&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, 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&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref name=&quot;:12&quot; /&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but not yet integrated into tested and validated methodologies&lt;ref name=&quot;:13&quot;&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;. Human validation stays notably necessary&lt;ref&gt;{{Cite journal|last=Alshami|first=Ahmad|last2=Elsayed|first2=Moustafa|last3=Ali|first3=Eslam|last4=Eltoukhy|first4=Abdelrahman E. E.|last5=Zayed|first5=Tarek|date=2023-07-09|title=Harnessing the Power of ChatGPT for Automating Systematic Review Process: Methodology, Case Study, Limitations, and Future Directions|url=https://www.mdpi.com/2079-8954/11/7/351|journal=Systems|language=en|volume=11|issue=7|pages=351|doi=10.3390/systems11070351|issn=2079-8954}}&lt;/ref&gt;,&lt;ref name=&quot;:13&quot; /&gt;. 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&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&amp;oldid=1360514388|journal=Wikipedia|language=en}}&lt;/ref&gt;, 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, and a &quot;collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation&quot;.&lt;ref&gt;{{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}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&amp;oldid=1360462340|journal=Wikipedia|language=en}}&lt;/ref&gt; &quot;A [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}&lt;/ref&gt; 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.&quot;&lt;ref&gt;{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}&lt;/ref&gt; Such graphs have a strong potential to conduct knowledge synthesis&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;&lt;ref name=&quot;:18&quot; /&gt;. They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;.  

Wikidata is multilingual... 

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]])&lt;ref&gt;{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&amp;oldid=1358552930|journal=Wikipedia|language=en}}&lt;/ref&gt;. Like all Wikimedia projects, Wikiversity is editable by everyone, have a discussion tab and a history log tab. 

=== Use of Wikidata and Wikiversity for research ===

== 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&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt; (e.g. the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;). 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)&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 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=&quot;wikitable&quot;
|+
!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 &quot;export to Wikidata QuickStatements&quot; 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|&quot;Merge&quot; 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. &lt;ref name=&quot;:5&quot; /&gt;). 

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=&quot;wikitable&quot;
|+
!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}}&lt;ref&gt;{{Cite journal|last=Paré|first=Guy|last2=Trudel|first2=Marie-Claude|last3=Jaana|first3=Mirou|last4=Kitsiou|first4=Spyros|date=2015-03|title=Synthesizing information systems knowledge: A typology of literature reviews|url=https://linkinghub.elsevier.com/retrieve/pii/S0378720614001116|journal=Information &amp; Management|language=en|volume=52|issue=2|pages=183–199|doi=10.1016/j.im.2014.08.008}}&lt;/ref&gt;. After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : 
{| class=&quot;wikitable&quot;
|+
!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 &quot;{{Wikidata entity link|Q137901202}}&quot; 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}}, &quot;energy transition&quot; 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 &quot;filter&quot; 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 :&lt;ref&gt;{{Cite journal|last=Nielsen|first=Finn Årup|last2=Lyhne|first2=Ivar|last3=Garigliotti|first3=Dario|last4=Butzbach|first4=Annika|last5=Ravn Boess|first5=Emilia|last6=Hose|first6=Katja|last7=Kørnøv|first7=Lone|date=2023|title=Environmental impact assessment reports in Wikidata and a Wikibase|url=https://repositum.tuwien.at/handle/20.500.12708/193492|language=en|publisher=CEUR-WS.org|volume=3443|pages=1–8|doi=10.34726/5421}}&lt;/ref&gt;)  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)&lt;ref name=&quot;:19&quot;&gt;Cañas, Alberto J., et al. &quot;CmapTools: A knowledge modeling and sharing environment.&quot; (2004): 125-135. https://thomaseskridge.com/assets/pdf/Canas-2004.pdf&lt;/ref&gt;. Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]&lt;ref name=&quot;:19&quot; /&gt;, Altas.Ti &quot;network&quot; feature...). The &quot;box and arrow&quot; 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”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. 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&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[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.&quot;&lt;ref name=&quot;:21&quot;&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; After identifying themes, qualitative researchers are expected to theorize the &quot;arrows&quot; between themes&lt;ref name=&quot;:21&quot; /&gt;R.esearchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;. 

=== 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 &quot;concept&quot; 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&lt;ref name=&quot;:22&quot; /&gt; 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...&lt;ref name=&quot;:22&quot;&gt;{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}&lt;/ref&gt;, and scholars have advised to take insight from philosophy to work on concepts&lt;ref&gt;{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}&lt;/ref&gt;. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt;. 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 &quot;something&quot; 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 &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, 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&amp;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 &quot;does not have effect&quot; seems contrary to what is listed in &quot;has goal&quot;.) We would however argue this is not a problem because &quot;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&quot;&lt;ref&gt;{{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}}&lt;/ref&gt;. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}&lt;/ref&gt;, as long as they are supported by references&lt;ref&gt;{{Cite journal|last=Amaral|first=Gabriel|last2=Piscopo|first2=Alessandro|last3=Kaffee|first3=Lucie-aimée|last4=Rodrigues|first4=Odinaldo|last5=Simperl|first5=Elena|date=2021-12-31|title=Assessing the Quality of Sources in Wikidata Across Languages: A Hybrid Approach|url=https://dl.acm.org/doi/10.1145/3484828|journal=Journal of Data and Information Quality|language=en|volume=13|issue=4|pages=1–35|doi=10.1145/3484828|issn=1936-1955}}&lt;/ref&gt;.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, the discrepancy between the stated goals of this movement and what it actually achieves is precisely what some authors are critiquing&lt;ref name=&quot;:20&quot; /&gt;.  

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 &quot;management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served&quot;&lt;ref&gt;{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research &amp; Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}&lt;/ref&gt;, &quot;choice&quot; 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 &quot;concept&quot; and &quot;political concept&quot;.]]
===== Concision =====
Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were &quot;in the tree of another value&quot;. 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&lt;nowiki/&gt;) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using &quot;instance of&quot; as transversal property and checking &quot;instance of or subclass of&quot;).

*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 &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;&lt;ref&gt;{{Cite journal|last=Vanegas-Cantarero|first=María M.|date=2020-12-01|title=Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries|url=https://www.wikidata.org/wiki/Q137901196|journal=Energy Research &amp; Social Science|language=English|volume=70|doi=10.1016/J.ERSS.2020.101716}}&lt;/ref&gt;. Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for &quot;final energy consumption&quot;. 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&lt;ref&gt;{{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}}&lt;/ref&gt;. 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 &quot;{{Wikidata entity link|Q127270577}}&quot;). 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 &quot;natures&quot; 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)&lt;ref&gt;Fleetwood, S. (2004). An ontology for organisation and management studies. ''Critical Realist Applications in Organisation and Management Studies'', 27–53.&lt;/ref&gt;. 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 &quot;supports multiple coexisting classification&quot; and allow multiple ontological frameworks to coexist.&lt;ref name=&quot;:8&quot; /&gt; 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=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{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}}&lt;/ref&gt; 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.

== 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 statements using a specitic source in order to map the content of a single academic article. First test here: https://w.wiki/PFqH
*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}}).

=== Properties visualisation ===

*Visualise the &quot;tree&quot; 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)

=== Concepts visualisation ===

*Map all statements related to a single item (e.g. [[d:Wikidata:Scholia|Scholia]] request &quot;topic in context&quot;)
*Map the statements that two items have in common to make comparisons.
[[File:Community energy and energy democracy.png|center|thumb|503x503px|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] .]]

=== Mapping sources consensus ===

*Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible.

== 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&lt;ref&gt;{{Cite journal|last=Maryl|first=Maciej|last2=Blaszczyńska|first2=Marta|last3=Zalotyńska|first3=Agnieszka|last4=Taylor|first4=Laurence|last5=Avanço|first5=Karla|last6=Balula|first6=Ana|last7=Buchner|first7=Anna|last8=Caliman|first8=Lorena|last9=Clivaz|first9=Claire|date=2021-01|title=Future of Scholarly Communication|url=https://hal.science/hal-03277615}}&lt;/ref&gt;). 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&lt;ref&gt;{{Cite web|url=https://eprints.whiterose.ac.uk/id/eprint/226771/|title=CLOUDSENSE Collaborative Review of Wikipedia Cloud Feedback article|last=Finney|first=D.|last2=Huang|first2=X.|date=2025-05-16|website=eprints.whiterose.ac.uk|language=en|doi=10.48785/100/329|access-date=2026-07-28|last3=Hinnie|first3=L.|last4=Sheppard|first4=N.}}&lt;/ref&gt;.

=== 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 &quot;remixed&quot; 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&lt;ref&gt;{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}&lt;/ref&gt;. Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to &quot;navigate&quot; 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 &quot;navigate&quot; 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&lt;ref name=&quot;:9&quot; /&gt;. 

=== Quality assessment ===
&lt;ref&gt;{{Citation|vauthors=((E. Peireira, A.))|year=2022|title=Usages et transformations d’un commun numérique : les données généalogiques équines|publisher=Paris 10|url=https://theses.fr/s357002|access-date=25 August 2026}}&lt;/ref&gt;  

=== Future research ===
- Can we &quot;reason&quot; 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 &lt;ref name=&quot;:24&quot;&gt;{{cite journal|date=June 2015|title=Novices’ Struggles with Conceptual and Theoretical Framing in Writing Dissertations and Papers for Publication|journal=Publications|publisher=Multidisciplinary Digital Publishing Institute|volume=3|issue=2|pages=104–119|doi=10.3390/publications3020104|issn=2304-6775|vauthors=((Casanave, C. P.)), ((Li, Y.))}}&lt;/ref&gt;. Conceptualization require to answer the question « what are our data examples of? » &lt;ref name=&quot;:24&quot; /&gt;.  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 &lt;ref name=&quot;:24&quot; /&gt;. 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_&amp;_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=&quot;wikitable sortable&quot;
! 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}}</text>
      <sha1>3wxt3eqmxscfdeu6ncsj997b6v60smp</sha1>
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    <revision>
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      <contributor>
        <username>Jeanne Noiraud</username>
        <id>1366702</id>
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      <comment>/* Concepts visualisation */ Adding scholia screenshot</comment>
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      <format>text/x-wiki</format>
      <text bytes="114300" sha1="kb84hrd6r0yc4ze2zpdg5g6c9e0crp6" xml:space="preserve">== Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) ==

== Acknowledgements ==
The present text was originally written on a Wikiversity page, if you are reading it in another format, you can find this page here : [[Just sustainability transitions: a living review|https://en.wikiversity.org/wiki/Just_sustainability_transitions:_a_living_review]]. You are free to add your comments on the page discussion section. 

=== Contributors ===
{| class=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-3-030-89460-3_2|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021|publisher=Springer International Publishing|isbn=978-3-030-89460-3|editor-last=Heffron|editor-first=Raphael J.|location=Cham|pages=9–19|language=en|doi=10.1007/978-3-030-89460-3_2}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.sciencedirect.com/science/article/pii/S0301421518302301|journal=Energy Policy|volume=119|pages=1–7|doi=10.1016/j.enpol.2018.04.014|issn=0301-4215}}&lt;/ref&gt;. 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&lt;ref name=&quot;:15&quot;&gt;{{Cite journal|date=2020-03-01|title=Micro-foundations of the multi-level perspective on socio-technical transitions: Developing a multi-dimensional model of agency through crossovers between social constructivism, evolutionary economics and neo-institutional theory|url=https://www.sciencedirect.com/science/article/abs/pii/S0040162518316111|journal=Technological Forecasting and Social Change|language=en-US|volume=152|pages=119894|doi=10.1016/j.techfore.2019.119894|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2023-08-01|title=A socio-technical transition perspective on positive tipping points in climate change mitigation: Analysing seven interacting feedback loops in offshore wind and electric vehicles acceleration|url=https://www.sciencedirect.com/science/article/pii/S0040162523003244|journal=Technological Forecasting and Social Change|language=en-US|volume=193|pages=122639|doi=10.1016/j.techfore.2023.122639|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Sovacool|first=Benjamin K.|last2=Geels|first2=Frank W.|last3=Andersen|first3=Allan Dahl|last4=Grubb|first4=Michael|last5=Jordan|first5=Andrew J.|last6=Kern|first6=Florian|last7=Kivimaa|first7=Paula|last8=Lockwood|first8=Matthew|last9=Markard|first9=Jochen|date=2025-03-01|title=The acceleration of low-carbon transitions: Insights, concepts, challenges, and new directions for research|url=https://www.sciencedirect.com/science/article/pii/S2214629625000295|journal=Energy Research &amp; Social Science|volume=121|pages=103948|doi=10.1016/j.erss.2025.103948|issn=2214-6296}}&lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Kepes|first=Sven|last2=Bennett|first2=Andrew A.|last3=McDaniel|first3=Michael A.|date=2014-09|title=Evidence-Based Management and the Trustworthiness of Our Cumulative Scientific Knowledge: Implications for Teaching, Research, and Practice|url=https://journals.aom.org/doi/10.5465/amle.2013.0193|journal=Academy of Management Learning &amp; Education|volume=13|issue=3|pages=446–466|doi=10.5465/amle.2013.0193|issn=1537-260X}}&lt;/ref&gt;&lt;ref&gt;Pfeffer, J., &amp; Sutton, R. I. (2006). Evidence-Based Management. Harvard Business Review, 13. &lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Bornmann|first=Lutz|last2=Mutz|first2=Rüdiger|date=2015|title=Growth rates of modern science: A bibliometric analysis based on the number of publications and cited references|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/asi.23329|journal=Journal of the Association for Information Science and Technology|language=en|volume=66|issue=11|pages=2215–2222|doi=10.1002/asi.23329|issn=2330-1643}}&lt;/ref&gt;&lt;ref name=&quot;:23&quot;&gt;{{Cite journal|last=Hanson|first=Mark A.|last2=Barreiro|first2=Pablo Gómez|last3=Crosetto|first3=Paolo|last4=Brockington|first4=Dan|date=2024-11-01|title=The strain on scientific publishing|url=https://doi.org/10.1162/qss_a_00327|journal=Quantitative Science Studies|volume=5|issue=4|pages=823–843|doi=10.1162/qss_a_00327|issn=2641-3337}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=http://blogs.nature.com/news/2014/05/global-scientific-output-doubles-every-nine-years.html|title=Global scientific output doubles every nine years : News blog|website=blogs.nature.com|language=en-US|access-date=2026-06-23}}&lt;/ref&gt;, 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&lt;ref name=&quot;:23&quot; /&gt;. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge&lt;ref&gt;{{Cite journal|last=Hartman|first=Yvonne|last2=Darab|first2=Sandy|date=2012-01-01|title=A Call for Slow Scholarship: A Case Study on the Intensification of Academic Life and Its Implications for Pedagogy|url=https://doi.org/10.1080/10714413.2012.643740|journal=Review of Education, Pedagogy, and Cultural Studies|volume=34|issue=1-2|pages=49–60|doi=10.1080/10714413.2012.643740|issn=1071-4413}}&lt;/ref&gt;. “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&lt;ref&gt;{{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)}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=L.|first=Mobley, David|last2=M.|first2=Zuckerman, Daniel|date=2015-02-03|title=A proposal for regularly updated review/survey articles: &quot;Perpetual Reviews&quot;|url=https://arxiv.org/abs/1502.01329|journal=arXiv.org|language=en|doi=10.48550/arXiv.1502.01329}}&lt;/ref&gt; and for shifting from static text format publications to dynamic knowledge mapping&lt;ref name=&quot;:11&quot;&gt;{{Cite web|url=https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/|title=The death of the literature review and the rise of the dynamic knowledge map - LSE Impact|last=Taster|date=2019-05-14|website=LSE Impact - Understanding impact and practice in academic research|access-date=2026-06-23}}&lt;/ref&gt;. This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : &lt;ref&gt;{{Cite journal|last=Elliott|first=Julian H.|last2=Synnot|first2=Anneliese|last3=Turner|first3=Tari|last4=Simmonds|first4=Mark|last5=Akl|first5=Elie A.|last6=McDonald|first6=Steve|last7=Salanti|first7=Georgia|last8=Meerpohl|first8=Joerg|last9=MacLehose|first9=Harriet|date=2017-11|title=Living systematic review: 1. Introduction—the why, what, when, and how|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435617306364|journal=Journal of Clinical Epidemiology|volume=91|pages=23–30|doi=10.1016/j.jclinepi.2017.08.010|issn=0895-4356}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Uttley|first=Lesley|last2=Quintana|first2=Daniel S.|last3=Montgomery|first3=Paul|last4=Carroll|first4=Christopher|last5=Page|first5=Matthew J.|last6=Falzon|first6=Louise|last7=Sutton|first7=Anthea|last8=Moher|first8=David|date=2023-04|title=The problems with systematic reviews: a living systematic review|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435623000112|journal=Journal of Clinical Epidemiology|volume=156|pages=30–41|doi=10.1016/j.jclinepi.2023.01.011|issn=0895-4356}}&lt;/ref&gt;,&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;). 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.

=== 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&lt;ref&gt;{{Cite journal|last=Haley|first=Usha C. V.|date=2023-09-01|title=Triviality and the Search for Scholarly Impact|url=https://doi.org/10.1177/01708406231175292|journal=Organization Studies|language=EN|volume=44|issue=9|pages=1547–1550|doi=10.1177/01708406231175292|issn=0170-8406}}&lt;/ref&gt;. Wikipedia is a key knowledge dissemination platform widely used by students&lt;ref&gt;{{Cite journal|last=Sunvy|first=Ahmed Shafkat|last2=Reza|first2=Raiyan Bin|date=2023-04-12|title=Students’ Perception of Wikipedia as an Academic Information Source|url=https://ejournal.undiksha.ac.id/index.php/IJERR/article/view/57572|journal=Indonesian Journal Of Educational Research and Review|volume=6|issue=1|pages=134–147|doi=10.23887/ijerr.v6i1.57572|issn=2621-8984}}&lt;/ref&gt; and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature&lt;ref&gt;{{Cite journal|last=Thompson|first=Neil|last2=Hanley|first2=Douglas|date=2017|title=Science Is Shaped by Wikipedia: Evidence from a Randomized Control Trial|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3039505|journal=SSRN Electronic Journal|doi=10.2139/ssrn.3039505|issn=1556-5068}}&lt;/ref&gt; and that some scholars cite directly Wikipedia&lt;ref&gt;{{Cite journal|last=Dooley|first=Patricia L.|date=2010-07-07|title=Wikipedia and the two-faced professoriate|url=https://doi.org/10.1145/1832772.1832803|journal=Proceedings of the 6th International Symposium on Wikis and Open Collaboration|series=WikiSym '10|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=1–2|doi=10.1145/1832772.1832803|isbn=978-1-4503-0056-8}}&lt;/ref&gt;. However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives&lt;ref&gt;{{Cite journal|last=Chen|first=Yan|last2=Farzan|first2=Rosta|last3=Kraut|first3=Robert|last4=YeckehZaare|first4=Iman|last5=Zhang|first5=Ark Fangzhou|date=2024-05|title=Motivating Experts to Contribute to Digital Public Goods: A Personalized Field Experiment on Wikipedia|url=https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4852|journal=Management Science|volume=70|issue=5|pages=3264–3280|doi=10.1287/mnsc.2023.4852|issn=0025-1909}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Kincaid|first=Dustin W.|last2=Beck|first2=Whitney S.|last3=Brandt|first3=Jessica E.|last4=Mars Brisbin|first4=Margaret|last5=Farrell|first5=Kaitlin J.|last6=Hondula|first6=Kelly L.|last7=Larson|first7=Erin I.|last8=Shogren|first8=Arial J.|date=2021|title=Wikipedia can help resolve information inequality in the aquatic sciences|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/lol2.10168|journal=Limnology and Oceanography Letters|language=en|volume=6|issue=1|pages=18–23|doi=10.1002/lol2.10168|issn=2378-2242}}&lt;/ref&gt;, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow&lt;ref name=&quot;:10&quot;&gt;Taraborelli, D., Mietchen, D., Alevizou, P., &amp; Gill, A. (2011, August). Expert participation on Wikipedia: Barriers and opportunities. Wikimania 2011, Haifa, Israel. &lt;nowiki&gt;http://upload.wikimedia.org/wikipedia/commons/4/4f/Expert_Participation_Survey_-_Wikimania_2011.pdf&lt;/nowiki&gt; &lt;/ref&gt;. In addition, expert participation is not immune to the gender gap&lt;ref name=&quot;:10&quot; /&gt;. Because of gender segregation in disciplines&lt;ref&gt;{{Cite journal|last=Ceci|first=Stephen J.|last2=Ginther|first2=Donna K.|last3=Kahn|first3=Shulamit|last4=Williams|first4=Wendy M.|date=2014-12-01|title=Women in Academic Science: A Changing Landscape|url=https://doi.org/10.1177/1529100614541236|journal=Psychological Science in the Public Interest|language=EN|volume=15|issue=3|pages=75–141|doi=10.1177/1529100614541236|issn=1529-1006}}&lt;/ref&gt;, this may be detrimental to the content coverage on “female” topics&lt;ref&gt;{{Cite journal|last=Lam|first=Shyong (Tony) K.|last2=Uduwage|first2=Anuradha|last3=Dong|first3=Zhenhua|last4=Sen|first4=Shilad|last5=Musicant|first5=David R.|last6=Terveen|first6=Loren|last7=Riedl|first7=John|date=2011-10-03|title=WP:clubhouse?: an exploration of Wikipedia's gender imbalance|url=https://dl.acm.org/doi/10.1145/2038558.2038560|language=en|publisher=ACM|pages=1–10|doi=10.1145/2038558.2038560|isbn=978-1-4503-0909-7}}&lt;/ref&gt;, 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 &quot;a fair and equitable process of moving towards a post-carbon society&quot;&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;. 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&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 

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&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:16&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; and fragmented results coming from various disciplines&lt;ref name=&quot;:20&quot;&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;. Then, making literature reviews &quot;living&quot; 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&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, 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&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref name=&quot;:12&quot; /&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but not yet integrated into tested and validated methodologies&lt;ref name=&quot;:13&quot;&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;. Human validation stays notably necessary&lt;ref&gt;{{Cite journal|last=Alshami|first=Ahmad|last2=Elsayed|first2=Moustafa|last3=Ali|first3=Eslam|last4=Eltoukhy|first4=Abdelrahman E. E.|last5=Zayed|first5=Tarek|date=2023-07-09|title=Harnessing the Power of ChatGPT for Automating Systematic Review Process: Methodology, Case Study, Limitations, and Future Directions|url=https://www.mdpi.com/2079-8954/11/7/351|journal=Systems|language=en|volume=11|issue=7|pages=351|doi=10.3390/systems11070351|issn=2079-8954}}&lt;/ref&gt;,&lt;ref name=&quot;:13&quot; /&gt;. 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&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&amp;oldid=1360514388|journal=Wikipedia|language=en}}&lt;/ref&gt;, 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, and a &quot;collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation&quot;.&lt;ref&gt;{{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}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&amp;oldid=1360462340|journal=Wikipedia|language=en}}&lt;/ref&gt; &quot;A [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}&lt;/ref&gt; 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.&quot;&lt;ref&gt;{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}&lt;/ref&gt; Such graphs have a strong potential to conduct knowledge synthesis&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;&lt;ref name=&quot;:18&quot; /&gt;. They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;.  

Wikidata is multilingual... 

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]])&lt;ref&gt;{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&amp;oldid=1358552930|journal=Wikipedia|language=en}}&lt;/ref&gt;. Like all Wikimedia projects, Wikiversity is editable by everyone, have a discussion tab and a history log tab. 

=== Use of Wikidata and Wikiversity for research ===

== 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&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt; (e.g. the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;). 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)&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 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=&quot;wikitable&quot;
|+
!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 &quot;export to Wikidata QuickStatements&quot; 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|&quot;Merge&quot; 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. &lt;ref name=&quot;:5&quot; /&gt;). 

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=&quot;wikitable&quot;
|+
!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}}&lt;ref&gt;{{Cite journal|last=Paré|first=Guy|last2=Trudel|first2=Marie-Claude|last3=Jaana|first3=Mirou|last4=Kitsiou|first4=Spyros|date=2015-03|title=Synthesizing information systems knowledge: A typology of literature reviews|url=https://linkinghub.elsevier.com/retrieve/pii/S0378720614001116|journal=Information &amp; Management|language=en|volume=52|issue=2|pages=183–199|doi=10.1016/j.im.2014.08.008}}&lt;/ref&gt;. After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : 
{| class=&quot;wikitable&quot;
|+
!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 &quot;{{Wikidata entity link|Q137901202}}&quot; 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}}, &quot;energy transition&quot; 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 &quot;filter&quot; 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 :&lt;ref&gt;{{Cite journal|last=Nielsen|first=Finn Årup|last2=Lyhne|first2=Ivar|last3=Garigliotti|first3=Dario|last4=Butzbach|first4=Annika|last5=Ravn Boess|first5=Emilia|last6=Hose|first6=Katja|last7=Kørnøv|first7=Lone|date=2023|title=Environmental impact assessment reports in Wikidata and a Wikibase|url=https://repositum.tuwien.at/handle/20.500.12708/193492|language=en|publisher=CEUR-WS.org|volume=3443|pages=1–8|doi=10.34726/5421}}&lt;/ref&gt;)  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)&lt;ref name=&quot;:19&quot;&gt;Cañas, Alberto J., et al. &quot;CmapTools: A knowledge modeling and sharing environment.&quot; (2004): 125-135. https://thomaseskridge.com/assets/pdf/Canas-2004.pdf&lt;/ref&gt;. Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]&lt;ref name=&quot;:19&quot; /&gt;, Altas.Ti &quot;network&quot; feature...). The &quot;box and arrow&quot; 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”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. 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&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[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.&quot;&lt;ref name=&quot;:21&quot;&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; After identifying themes, qualitative researchers are expected to theorize the &quot;arrows&quot; between themes&lt;ref name=&quot;:21&quot; /&gt;R.esearchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;. 

=== 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 &quot;concept&quot; 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&lt;ref name=&quot;:22&quot; /&gt; 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...&lt;ref name=&quot;:22&quot;&gt;{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}&lt;/ref&gt;, and scholars have advised to take insight from philosophy to work on concepts&lt;ref&gt;{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}&lt;/ref&gt;. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt;. 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 &quot;something&quot; 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 &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, 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&amp;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 &quot;does not have effect&quot; seems contrary to what is listed in &quot;has goal&quot;.) We would however argue this is not a problem because &quot;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&quot;&lt;ref&gt;{{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}}&lt;/ref&gt;. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}&lt;/ref&gt;, as long as they are supported by references&lt;ref&gt;{{Cite journal|last=Amaral|first=Gabriel|last2=Piscopo|first2=Alessandro|last3=Kaffee|first3=Lucie-aimée|last4=Rodrigues|first4=Odinaldo|last5=Simperl|first5=Elena|date=2021-12-31|title=Assessing the Quality of Sources in Wikidata Across Languages: A Hybrid Approach|url=https://dl.acm.org/doi/10.1145/3484828|journal=Journal of Data and Information Quality|language=en|volume=13|issue=4|pages=1–35|doi=10.1145/3484828|issn=1936-1955}}&lt;/ref&gt;.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, the discrepancy between the stated goals of this movement and what it actually achieves is precisely what some authors are critiquing&lt;ref name=&quot;:20&quot; /&gt;.  

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 &quot;management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served&quot;&lt;ref&gt;{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research &amp; Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}&lt;/ref&gt;, &quot;choice&quot; 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 &quot;concept&quot; and &quot;political concept&quot;.]]
===== Concision =====
Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were &quot;in the tree of another value&quot;. 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&lt;nowiki/&gt;) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using &quot;instance of&quot; as transversal property and checking &quot;instance of or subclass of&quot;).

*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 &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;&lt;ref&gt;{{Cite journal|last=Vanegas-Cantarero|first=María M.|date=2020-12-01|title=Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries|url=https://www.wikidata.org/wiki/Q137901196|journal=Energy Research &amp; Social Science|language=English|volume=70|doi=10.1016/J.ERSS.2020.101716}}&lt;/ref&gt;. Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for &quot;final energy consumption&quot;. 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&lt;ref&gt;{{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}}&lt;/ref&gt;. 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 &quot;{{Wikidata entity link|Q127270577}}&quot;). 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 &quot;natures&quot; 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)&lt;ref&gt;Fleetwood, S. (2004). An ontology for organisation and management studies. ''Critical Realist Applications in Organisation and Management Studies'', 27–53.&lt;/ref&gt;. 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 &quot;supports multiple coexisting classification&quot; and allow multiple ontological frameworks to coexist.&lt;ref name=&quot;:8&quot; /&gt; 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=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{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}}&lt;/ref&gt; 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.

== 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 statements using a specitic source in order to map the content of a single academic article. First test here: https://w.wiki/PFqH
*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}}).

=== Properties visualisation ===

*Visualise the &quot;tree&quot; 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)

=== Concepts visualisation ===

*Map all statements related to a single item (e.g. [[d:Wikidata:Scholia|Scholia]] request &quot;topic in context&quot;)[[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 &quot;The topic in context&quot; 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] .]]

=== Mapping sources consensus ===

*Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible.

== 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&lt;ref&gt;{{Cite journal|last=Maryl|first=Maciej|last2=Blaszczyńska|first2=Marta|last3=Zalotyńska|first3=Agnieszka|last4=Taylor|first4=Laurence|last5=Avanço|first5=Karla|last6=Balula|first6=Ana|last7=Buchner|first7=Anna|last8=Caliman|first8=Lorena|last9=Clivaz|first9=Claire|date=2021-01|title=Future of Scholarly Communication|url=https://hal.science/hal-03277615}}&lt;/ref&gt;). 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&lt;ref&gt;{{Cite web|url=https://eprints.whiterose.ac.uk/id/eprint/226771/|title=CLOUDSENSE Collaborative Review of Wikipedia Cloud Feedback article|last=Finney|first=D.|last2=Huang|first2=X.|date=2025-05-16|website=eprints.whiterose.ac.uk|language=en|doi=10.48785/100/329|access-date=2026-07-28|last3=Hinnie|first3=L.|last4=Sheppard|first4=N.}}&lt;/ref&gt;.

=== 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 &quot;remixed&quot; 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&lt;ref&gt;{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}&lt;/ref&gt;. Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to &quot;navigate&quot; 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 &quot;navigate&quot; 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&lt;ref name=&quot;:9&quot; /&gt;. 

=== Quality assessment ===
&lt;ref&gt;{{Citation|vauthors=((E. Peireira, A.))|year=2022|title=Usages et transformations d’un commun numérique : les données généalogiques équines|publisher=Paris 10|url=https://theses.fr/s357002|access-date=25 August 2026}}&lt;/ref&gt;  

=== Future research ===
- Can we &quot;reason&quot; 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 &lt;ref name=&quot;:24&quot;&gt;{{cite journal|date=June 2015|title=Novices’ Struggles with Conceptual and Theoretical Framing in Writing Dissertations and Papers for Publication|journal=Publications|publisher=Multidisciplinary Digital Publishing Institute|volume=3|issue=2|pages=104–119|doi=10.3390/publications3020104|issn=2304-6775|vauthors=((Casanave, C. P.)), ((Li, Y.))}}&lt;/ref&gt;. Conceptualization require to answer the question « what are our data examples of? » &lt;ref name=&quot;:24&quot; /&gt;.  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 &lt;ref name=&quot;:24&quot; /&gt;. 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_&amp;_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=&quot;wikitable sortable&quot;
! 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}}</text>
      <sha1>kb84hrd6r0yc4ze2zpdg5g6c9e0crp6</sha1>
    </revision>
    <revision>
      <id>2829729</id>
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      <timestamp>2026-08-30T14:25:00Z</timestamp>
      <contributor>
        <username>Jeanne Noiraud</username>
        <id>1366702</id>
      </contributor>
      <comment>/* Properties visualisation */ Adding example of visualisation with Wikidata Graph Builder</comment>
      <origin>2829729</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="114638" sha1="d93miz37bu1czoo2mwzzh0syrna0055" xml:space="preserve">== Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) ==

== Acknowledgements ==
The present text was originally written on a Wikiversity page, if you are reading it in another format, you can find this page here : [[Just sustainability transitions: a living review|https://en.wikiversity.org/wiki/Just_sustainability_transitions:_a_living_review]]. You are free to add your comments on the page discussion section. 

=== Contributors ===
{| class=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-3-030-89460-3_2|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021|publisher=Springer International Publishing|isbn=978-3-030-89460-3|editor-last=Heffron|editor-first=Raphael J.|location=Cham|pages=9–19|language=en|doi=10.1007/978-3-030-89460-3_2}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.sciencedirect.com/science/article/pii/S0301421518302301|journal=Energy Policy|volume=119|pages=1–7|doi=10.1016/j.enpol.2018.04.014|issn=0301-4215}}&lt;/ref&gt;. 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&lt;ref name=&quot;:15&quot;&gt;{{Cite journal|date=2020-03-01|title=Micro-foundations of the multi-level perspective on socio-technical transitions: Developing a multi-dimensional model of agency through crossovers between social constructivism, evolutionary economics and neo-institutional theory|url=https://www.sciencedirect.com/science/article/abs/pii/S0040162518316111|journal=Technological Forecasting and Social Change|language=en-US|volume=152|pages=119894|doi=10.1016/j.techfore.2019.119894|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2023-08-01|title=A socio-technical transition perspective on positive tipping points in climate change mitigation: Analysing seven interacting feedback loops in offshore wind and electric vehicles acceleration|url=https://www.sciencedirect.com/science/article/pii/S0040162523003244|journal=Technological Forecasting and Social Change|language=en-US|volume=193|pages=122639|doi=10.1016/j.techfore.2023.122639|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Sovacool|first=Benjamin K.|last2=Geels|first2=Frank W.|last3=Andersen|first3=Allan Dahl|last4=Grubb|first4=Michael|last5=Jordan|first5=Andrew J.|last6=Kern|first6=Florian|last7=Kivimaa|first7=Paula|last8=Lockwood|first8=Matthew|last9=Markard|first9=Jochen|date=2025-03-01|title=The acceleration of low-carbon transitions: Insights, concepts, challenges, and new directions for research|url=https://www.sciencedirect.com/science/article/pii/S2214629625000295|journal=Energy Research &amp; Social Science|volume=121|pages=103948|doi=10.1016/j.erss.2025.103948|issn=2214-6296}}&lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Kepes|first=Sven|last2=Bennett|first2=Andrew A.|last3=McDaniel|first3=Michael A.|date=2014-09|title=Evidence-Based Management and the Trustworthiness of Our Cumulative Scientific Knowledge: Implications for Teaching, Research, and Practice|url=https://journals.aom.org/doi/10.5465/amle.2013.0193|journal=Academy of Management Learning &amp; Education|volume=13|issue=3|pages=446–466|doi=10.5465/amle.2013.0193|issn=1537-260X}}&lt;/ref&gt;&lt;ref&gt;Pfeffer, J., &amp; Sutton, R. I. (2006). Evidence-Based Management. Harvard Business Review, 13. &lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Bornmann|first=Lutz|last2=Mutz|first2=Rüdiger|date=2015|title=Growth rates of modern science: A bibliometric analysis based on the number of publications and cited references|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/asi.23329|journal=Journal of the Association for Information Science and Technology|language=en|volume=66|issue=11|pages=2215–2222|doi=10.1002/asi.23329|issn=2330-1643}}&lt;/ref&gt;&lt;ref name=&quot;:23&quot;&gt;{{Cite journal|last=Hanson|first=Mark A.|last2=Barreiro|first2=Pablo Gómez|last3=Crosetto|first3=Paolo|last4=Brockington|first4=Dan|date=2024-11-01|title=The strain on scientific publishing|url=https://doi.org/10.1162/qss_a_00327|journal=Quantitative Science Studies|volume=5|issue=4|pages=823–843|doi=10.1162/qss_a_00327|issn=2641-3337}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=http://blogs.nature.com/news/2014/05/global-scientific-output-doubles-every-nine-years.html|title=Global scientific output doubles every nine years : News blog|website=blogs.nature.com|language=en-US|access-date=2026-06-23}}&lt;/ref&gt;, 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&lt;ref name=&quot;:23&quot; /&gt;. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge&lt;ref&gt;{{Cite journal|last=Hartman|first=Yvonne|last2=Darab|first2=Sandy|date=2012-01-01|title=A Call for Slow Scholarship: A Case Study on the Intensification of Academic Life and Its Implications for Pedagogy|url=https://doi.org/10.1080/10714413.2012.643740|journal=Review of Education, Pedagogy, and Cultural Studies|volume=34|issue=1-2|pages=49–60|doi=10.1080/10714413.2012.643740|issn=1071-4413}}&lt;/ref&gt;. “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&lt;ref&gt;{{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)}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=L.|first=Mobley, David|last2=M.|first2=Zuckerman, Daniel|date=2015-02-03|title=A proposal for regularly updated review/survey articles: &quot;Perpetual Reviews&quot;|url=https://arxiv.org/abs/1502.01329|journal=arXiv.org|language=en|doi=10.48550/arXiv.1502.01329}}&lt;/ref&gt; and for shifting from static text format publications to dynamic knowledge mapping&lt;ref name=&quot;:11&quot;&gt;{{Cite web|url=https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/|title=The death of the literature review and the rise of the dynamic knowledge map - LSE Impact|last=Taster|date=2019-05-14|website=LSE Impact - Understanding impact and practice in academic research|access-date=2026-06-23}}&lt;/ref&gt;. This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : &lt;ref&gt;{{Cite journal|last=Elliott|first=Julian H.|last2=Synnot|first2=Anneliese|last3=Turner|first3=Tari|last4=Simmonds|first4=Mark|last5=Akl|first5=Elie A.|last6=McDonald|first6=Steve|last7=Salanti|first7=Georgia|last8=Meerpohl|first8=Joerg|last9=MacLehose|first9=Harriet|date=2017-11|title=Living systematic review: 1. Introduction—the why, what, when, and how|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435617306364|journal=Journal of Clinical Epidemiology|volume=91|pages=23–30|doi=10.1016/j.jclinepi.2017.08.010|issn=0895-4356}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Uttley|first=Lesley|last2=Quintana|first2=Daniel S.|last3=Montgomery|first3=Paul|last4=Carroll|first4=Christopher|last5=Page|first5=Matthew J.|last6=Falzon|first6=Louise|last7=Sutton|first7=Anthea|last8=Moher|first8=David|date=2023-04|title=The problems with systematic reviews: a living systematic review|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435623000112|journal=Journal of Clinical Epidemiology|volume=156|pages=30–41|doi=10.1016/j.jclinepi.2023.01.011|issn=0895-4356}}&lt;/ref&gt;,&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;). 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.

=== 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&lt;ref&gt;{{Cite journal|last=Haley|first=Usha C. V.|date=2023-09-01|title=Triviality and the Search for Scholarly Impact|url=https://doi.org/10.1177/01708406231175292|journal=Organization Studies|language=EN|volume=44|issue=9|pages=1547–1550|doi=10.1177/01708406231175292|issn=0170-8406}}&lt;/ref&gt;. Wikipedia is a key knowledge dissemination platform widely used by students&lt;ref&gt;{{Cite journal|last=Sunvy|first=Ahmed Shafkat|last2=Reza|first2=Raiyan Bin|date=2023-04-12|title=Students’ Perception of Wikipedia as an Academic Information Source|url=https://ejournal.undiksha.ac.id/index.php/IJERR/article/view/57572|journal=Indonesian Journal Of Educational Research and Review|volume=6|issue=1|pages=134–147|doi=10.23887/ijerr.v6i1.57572|issn=2621-8984}}&lt;/ref&gt; and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature&lt;ref&gt;{{Cite journal|last=Thompson|first=Neil|last2=Hanley|first2=Douglas|date=2017|title=Science Is Shaped by Wikipedia: Evidence from a Randomized Control Trial|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3039505|journal=SSRN Electronic Journal|doi=10.2139/ssrn.3039505|issn=1556-5068}}&lt;/ref&gt; and that some scholars cite directly Wikipedia&lt;ref&gt;{{Cite journal|last=Dooley|first=Patricia L.|date=2010-07-07|title=Wikipedia and the two-faced professoriate|url=https://doi.org/10.1145/1832772.1832803|journal=Proceedings of the 6th International Symposium on Wikis and Open Collaboration|series=WikiSym '10|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=1–2|doi=10.1145/1832772.1832803|isbn=978-1-4503-0056-8}}&lt;/ref&gt;. However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives&lt;ref&gt;{{Cite journal|last=Chen|first=Yan|last2=Farzan|first2=Rosta|last3=Kraut|first3=Robert|last4=YeckehZaare|first4=Iman|last5=Zhang|first5=Ark Fangzhou|date=2024-05|title=Motivating Experts to Contribute to Digital Public Goods: A Personalized Field Experiment on Wikipedia|url=https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4852|journal=Management Science|volume=70|issue=5|pages=3264–3280|doi=10.1287/mnsc.2023.4852|issn=0025-1909}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Kincaid|first=Dustin W.|last2=Beck|first2=Whitney S.|last3=Brandt|first3=Jessica E.|last4=Mars Brisbin|first4=Margaret|last5=Farrell|first5=Kaitlin J.|last6=Hondula|first6=Kelly L.|last7=Larson|first7=Erin I.|last8=Shogren|first8=Arial J.|date=2021|title=Wikipedia can help resolve information inequality in the aquatic sciences|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/lol2.10168|journal=Limnology and Oceanography Letters|language=en|volume=6|issue=1|pages=18–23|doi=10.1002/lol2.10168|issn=2378-2242}}&lt;/ref&gt;, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow&lt;ref name=&quot;:10&quot;&gt;Taraborelli, D., Mietchen, D., Alevizou, P., &amp; Gill, A. (2011, August). Expert participation on Wikipedia: Barriers and opportunities. Wikimania 2011, Haifa, Israel. &lt;nowiki&gt;http://upload.wikimedia.org/wikipedia/commons/4/4f/Expert_Participation_Survey_-_Wikimania_2011.pdf&lt;/nowiki&gt; &lt;/ref&gt;. In addition, expert participation is not immune to the gender gap&lt;ref name=&quot;:10&quot; /&gt;. Because of gender segregation in disciplines&lt;ref&gt;{{Cite journal|last=Ceci|first=Stephen J.|last2=Ginther|first2=Donna K.|last3=Kahn|first3=Shulamit|last4=Williams|first4=Wendy M.|date=2014-12-01|title=Women in Academic Science: A Changing Landscape|url=https://doi.org/10.1177/1529100614541236|journal=Psychological Science in the Public Interest|language=EN|volume=15|issue=3|pages=75–141|doi=10.1177/1529100614541236|issn=1529-1006}}&lt;/ref&gt;, this may be detrimental to the content coverage on “female” topics&lt;ref&gt;{{Cite journal|last=Lam|first=Shyong (Tony) K.|last2=Uduwage|first2=Anuradha|last3=Dong|first3=Zhenhua|last4=Sen|first4=Shilad|last5=Musicant|first5=David R.|last6=Terveen|first6=Loren|last7=Riedl|first7=John|date=2011-10-03|title=WP:clubhouse?: an exploration of Wikipedia's gender imbalance|url=https://dl.acm.org/doi/10.1145/2038558.2038560|language=en|publisher=ACM|pages=1–10|doi=10.1145/2038558.2038560|isbn=978-1-4503-0909-7}}&lt;/ref&gt;, 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 &quot;a fair and equitable process of moving towards a post-carbon society&quot;&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;. 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&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 

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&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:16&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; and fragmented results coming from various disciplines&lt;ref name=&quot;:20&quot;&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;. Then, making literature reviews &quot;living&quot; 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&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, 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&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref name=&quot;:12&quot; /&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but not yet integrated into tested and validated methodologies&lt;ref name=&quot;:13&quot;&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;. Human validation stays notably necessary&lt;ref&gt;{{Cite journal|last=Alshami|first=Ahmad|last2=Elsayed|first2=Moustafa|last3=Ali|first3=Eslam|last4=Eltoukhy|first4=Abdelrahman E. E.|last5=Zayed|first5=Tarek|date=2023-07-09|title=Harnessing the Power of ChatGPT for Automating Systematic Review Process: Methodology, Case Study, Limitations, and Future Directions|url=https://www.mdpi.com/2079-8954/11/7/351|journal=Systems|language=en|volume=11|issue=7|pages=351|doi=10.3390/systems11070351|issn=2079-8954}}&lt;/ref&gt;,&lt;ref name=&quot;:13&quot; /&gt;. 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&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&amp;oldid=1360514388|journal=Wikipedia|language=en}}&lt;/ref&gt;, 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, and a &quot;collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation&quot;.&lt;ref&gt;{{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}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&amp;oldid=1360462340|journal=Wikipedia|language=en}}&lt;/ref&gt; &quot;A [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}&lt;/ref&gt; 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.&quot;&lt;ref&gt;{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}&lt;/ref&gt; Such graphs have a strong potential to conduct knowledge synthesis&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;&lt;ref name=&quot;:18&quot; /&gt;. They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;.  

Wikidata is multilingual... 

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]])&lt;ref&gt;{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&amp;oldid=1358552930|journal=Wikipedia|language=en}}&lt;/ref&gt;. Like all Wikimedia projects, Wikiversity is editable by everyone, have a discussion tab and a history log tab. 

=== Use of Wikidata and Wikiversity for research ===

== 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&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt; (e.g. the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;). 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)&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 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=&quot;wikitable&quot;
|+
!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 &quot;export to Wikidata QuickStatements&quot; 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|&quot;Merge&quot; 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. &lt;ref name=&quot;:5&quot; /&gt;). 

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=&quot;wikitable&quot;
|+
!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}}&lt;ref&gt;{{Cite journal|last=Paré|first=Guy|last2=Trudel|first2=Marie-Claude|last3=Jaana|first3=Mirou|last4=Kitsiou|first4=Spyros|date=2015-03|title=Synthesizing information systems knowledge: A typology of literature reviews|url=https://linkinghub.elsevier.com/retrieve/pii/S0378720614001116|journal=Information &amp; Management|language=en|volume=52|issue=2|pages=183–199|doi=10.1016/j.im.2014.08.008}}&lt;/ref&gt;. After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : 
{| class=&quot;wikitable&quot;
|+
!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 &quot;{{Wikidata entity link|Q137901202}}&quot; 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}}, &quot;energy transition&quot; 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 &quot;filter&quot; 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 :&lt;ref&gt;{{Cite journal|last=Nielsen|first=Finn Årup|last2=Lyhne|first2=Ivar|last3=Garigliotti|first3=Dario|last4=Butzbach|first4=Annika|last5=Ravn Boess|first5=Emilia|last6=Hose|first6=Katja|last7=Kørnøv|first7=Lone|date=2023|title=Environmental impact assessment reports in Wikidata and a Wikibase|url=https://repositum.tuwien.at/handle/20.500.12708/193492|language=en|publisher=CEUR-WS.org|volume=3443|pages=1–8|doi=10.34726/5421}}&lt;/ref&gt;)  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)&lt;ref name=&quot;:19&quot;&gt;Cañas, Alberto J., et al. &quot;CmapTools: A knowledge modeling and sharing environment.&quot; (2004): 125-135. https://thomaseskridge.com/assets/pdf/Canas-2004.pdf&lt;/ref&gt;. Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]&lt;ref name=&quot;:19&quot; /&gt;, Altas.Ti &quot;network&quot; feature...). The &quot;box and arrow&quot; 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”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. 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&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[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.&quot;&lt;ref name=&quot;:21&quot;&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; After identifying themes, qualitative researchers are expected to theorize the &quot;arrows&quot; between themes&lt;ref name=&quot;:21&quot; /&gt;R.esearchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;. 

=== 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 &quot;concept&quot; 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&lt;ref name=&quot;:22&quot; /&gt; 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...&lt;ref name=&quot;:22&quot;&gt;{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}&lt;/ref&gt;, and scholars have advised to take insight from philosophy to work on concepts&lt;ref&gt;{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}&lt;/ref&gt;. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt;. 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 &quot;something&quot; 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 &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, 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&amp;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 &quot;does not have effect&quot; seems contrary to what is listed in &quot;has goal&quot;.) We would however argue this is not a problem because &quot;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&quot;&lt;ref&gt;{{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}}&lt;/ref&gt;. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}&lt;/ref&gt;, as long as they are supported by references&lt;ref&gt;{{Cite journal|last=Amaral|first=Gabriel|last2=Piscopo|first2=Alessandro|last3=Kaffee|first3=Lucie-aimée|last4=Rodrigues|first4=Odinaldo|last5=Simperl|first5=Elena|date=2021-12-31|title=Assessing the Quality of Sources in Wikidata Across Languages: A Hybrid Approach|url=https://dl.acm.org/doi/10.1145/3484828|journal=Journal of Data and Information Quality|language=en|volume=13|issue=4|pages=1–35|doi=10.1145/3484828|issn=1936-1955}}&lt;/ref&gt;.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, the discrepancy between the stated goals of this movement and what it actually achieves is precisely what some authors are critiquing&lt;ref name=&quot;:20&quot; /&gt;.  

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 &quot;management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served&quot;&lt;ref&gt;{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research &amp; Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}&lt;/ref&gt;, &quot;choice&quot; 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 &quot;concept&quot; and &quot;political concept&quot;.]]
===== Concision =====
Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were &quot;in the tree of another value&quot;. 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&lt;nowiki/&gt;) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using &quot;instance of&quot; as transversal property and checking &quot;instance of or subclass of&quot;).

*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 &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;&lt;ref&gt;{{Cite journal|last=Vanegas-Cantarero|first=María M.|date=2020-12-01|title=Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries|url=https://www.wikidata.org/wiki/Q137901196|journal=Energy Research &amp; Social Science|language=English|volume=70|doi=10.1016/J.ERSS.2020.101716}}&lt;/ref&gt;. Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for &quot;final energy consumption&quot;. 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&lt;ref&gt;{{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}}&lt;/ref&gt;. 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 &quot;{{Wikidata entity link|Q127270577}}&quot;). 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 &quot;natures&quot; 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)&lt;ref&gt;Fleetwood, S. (2004). An ontology for organisation and management studies. ''Critical Realist Applications in Organisation and Management Studies'', 27–53.&lt;/ref&gt;. 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 &quot;supports multiple coexisting classification&quot; and allow multiple ontological frameworks to coexist.&lt;ref name=&quot;:8&quot; /&gt; 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=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{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}}&lt;/ref&gt; 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.

== 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 statements using a specitic source in order to map the content of a single academic article. First test here: https://w.wiki/PFqH
*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}}).

=== Properties visualisation ===

*Visualise the &quot;tree&quot; 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 &quot;instance of&quot; values for energy democracy in Wikidata.png|center|thumb|553x553px|Visualisation of the &quot;{{Wikidata entity link|P31}}&quot; 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 &quot;topic in context&quot;)[[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 &quot;The topic in context&quot; 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] .]]

=== Mapping sources consensus ===

*Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible.

== 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&lt;ref&gt;{{Cite journal|last=Maryl|first=Maciej|last2=Blaszczyńska|first2=Marta|last3=Zalotyńska|first3=Agnieszka|last4=Taylor|first4=Laurence|last5=Avanço|first5=Karla|last6=Balula|first6=Ana|last7=Buchner|first7=Anna|last8=Caliman|first8=Lorena|last9=Clivaz|first9=Claire|date=2021-01|title=Future of Scholarly Communication|url=https://hal.science/hal-03277615}}&lt;/ref&gt;). 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&lt;ref&gt;{{Cite web|url=https://eprints.whiterose.ac.uk/id/eprint/226771/|title=CLOUDSENSE Collaborative Review of Wikipedia Cloud Feedback article|last=Finney|first=D.|last2=Huang|first2=X.|date=2025-05-16|website=eprints.whiterose.ac.uk|language=en|doi=10.48785/100/329|access-date=2026-07-28|last3=Hinnie|first3=L.|last4=Sheppard|first4=N.}}&lt;/ref&gt;.

=== 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 &quot;remixed&quot; 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&lt;ref&gt;{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}&lt;/ref&gt;. Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to &quot;navigate&quot; 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 &quot;navigate&quot; 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&lt;ref name=&quot;:9&quot; /&gt;. 

=== Quality assessment ===
&lt;ref&gt;{{Citation|vauthors=((E. Peireira, A.))|year=2022|title=Usages et transformations d’un commun numérique : les données généalogiques équines|publisher=Paris 10|url=https://theses.fr/s357002|access-date=25 August 2026}}&lt;/ref&gt;  

=== Future research ===
- Can we &quot;reason&quot; 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 &lt;ref name=&quot;:24&quot;&gt;{{cite journal|date=June 2015|title=Novices’ Struggles with Conceptual and Theoretical Framing in Writing Dissertations and Papers for Publication|journal=Publications|publisher=Multidisciplinary Digital Publishing Institute|volume=3|issue=2|pages=104–119|doi=10.3390/publications3020104|issn=2304-6775|vauthors=((Casanave, C. P.)), ((Li, Y.))}}&lt;/ref&gt;. Conceptualization require to answer the question « what are our data examples of? » &lt;ref name=&quot;:24&quot; /&gt;.  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 &lt;ref name=&quot;:24&quot; /&gt;. 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_&amp;_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=&quot;wikitable sortable&quot;
! 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
|-
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|}


== References ==
{{References}}</text>
      <sha1>d93miz37bu1czoo2mwzzh0syrna0055</sha1>
    </revision>
    <revision>
      <id>2829769</id>
      <parentid>2829729</parentid>
      <timestamp>2026-08-30T20:11:28Z</timestamp>
      <contributor>
        <username>Jeanne Noiraud</username>
        <id>1366702</id>
      </contributor>
      <comment>/* Knowledge modelling in Wikidata : second round of analysis */</comment>
      <origin>2829769</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="114786" sha1="0nq5n3wkqxgn9eo6mbtxdg9z16j3elj" xml:space="preserve">== Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) ==

== Acknowledgements ==
The present text was originally written on a Wikiversity page, if you are reading it in another format, you can find this page here : [[Just sustainability transitions: a living review|https://en.wikiversity.org/wiki/Just_sustainability_transitions:_a_living_review]]. You are free to add your comments on the page discussion section. 

=== Contributors ===
{| class=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-3-030-89460-3_2|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021|publisher=Springer International Publishing|isbn=978-3-030-89460-3|editor-last=Heffron|editor-first=Raphael J.|location=Cham|pages=9–19|language=en|doi=10.1007/978-3-030-89460-3_2}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.sciencedirect.com/science/article/pii/S0301421518302301|journal=Energy Policy|volume=119|pages=1–7|doi=10.1016/j.enpol.2018.04.014|issn=0301-4215}}&lt;/ref&gt;. 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&lt;ref name=&quot;:15&quot;&gt;{{Cite journal|date=2020-03-01|title=Micro-foundations of the multi-level perspective on socio-technical transitions: Developing a multi-dimensional model of agency through crossovers between social constructivism, evolutionary economics and neo-institutional theory|url=https://www.sciencedirect.com/science/article/abs/pii/S0040162518316111|journal=Technological Forecasting and Social Change|language=en-US|volume=152|pages=119894|doi=10.1016/j.techfore.2019.119894|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2023-08-01|title=A socio-technical transition perspective on positive tipping points in climate change mitigation: Analysing seven interacting feedback loops in offshore wind and electric vehicles acceleration|url=https://www.sciencedirect.com/science/article/pii/S0040162523003244|journal=Technological Forecasting and Social Change|language=en-US|volume=193|pages=122639|doi=10.1016/j.techfore.2023.122639|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Sovacool|first=Benjamin K.|last2=Geels|first2=Frank W.|last3=Andersen|first3=Allan Dahl|last4=Grubb|first4=Michael|last5=Jordan|first5=Andrew J.|last6=Kern|first6=Florian|last7=Kivimaa|first7=Paula|last8=Lockwood|first8=Matthew|last9=Markard|first9=Jochen|date=2025-03-01|title=The acceleration of low-carbon transitions: Insights, concepts, challenges, and new directions for research|url=https://www.sciencedirect.com/science/article/pii/S2214629625000295|journal=Energy Research &amp; Social Science|volume=121|pages=103948|doi=10.1016/j.erss.2025.103948|issn=2214-6296}}&lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Kepes|first=Sven|last2=Bennett|first2=Andrew A.|last3=McDaniel|first3=Michael A.|date=2014-09|title=Evidence-Based Management and the Trustworthiness of Our Cumulative Scientific Knowledge: Implications for Teaching, Research, and Practice|url=https://journals.aom.org/doi/10.5465/amle.2013.0193|journal=Academy of Management Learning &amp; Education|volume=13|issue=3|pages=446–466|doi=10.5465/amle.2013.0193|issn=1537-260X}}&lt;/ref&gt;&lt;ref&gt;Pfeffer, J., &amp; Sutton, R. I. (2006). Evidence-Based Management. Harvard Business Review, 13. &lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Bornmann|first=Lutz|last2=Mutz|first2=Rüdiger|date=2015|title=Growth rates of modern science: A bibliometric analysis based on the number of publications and cited references|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/asi.23329|journal=Journal of the Association for Information Science and Technology|language=en|volume=66|issue=11|pages=2215–2222|doi=10.1002/asi.23329|issn=2330-1643}}&lt;/ref&gt;&lt;ref name=&quot;:23&quot;&gt;{{Cite journal|last=Hanson|first=Mark A.|last2=Barreiro|first2=Pablo Gómez|last3=Crosetto|first3=Paolo|last4=Brockington|first4=Dan|date=2024-11-01|title=The strain on scientific publishing|url=https://doi.org/10.1162/qss_a_00327|journal=Quantitative Science Studies|volume=5|issue=4|pages=823–843|doi=10.1162/qss_a_00327|issn=2641-3337}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=http://blogs.nature.com/news/2014/05/global-scientific-output-doubles-every-nine-years.html|title=Global scientific output doubles every nine years : News blog|website=blogs.nature.com|language=en-US|access-date=2026-06-23}}&lt;/ref&gt;, 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&lt;ref name=&quot;:23&quot; /&gt;. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge&lt;ref&gt;{{Cite journal|last=Hartman|first=Yvonne|last2=Darab|first2=Sandy|date=2012-01-01|title=A Call for Slow Scholarship: A Case Study on the Intensification of Academic Life and Its Implications for Pedagogy|url=https://doi.org/10.1080/10714413.2012.643740|journal=Review of Education, Pedagogy, and Cultural Studies|volume=34|issue=1-2|pages=49–60|doi=10.1080/10714413.2012.643740|issn=1071-4413}}&lt;/ref&gt;. “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&lt;ref&gt;{{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)}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=L.|first=Mobley, David|last2=M.|first2=Zuckerman, Daniel|date=2015-02-03|title=A proposal for regularly updated review/survey articles: &quot;Perpetual Reviews&quot;|url=https://arxiv.org/abs/1502.01329|journal=arXiv.org|language=en|doi=10.48550/arXiv.1502.01329}}&lt;/ref&gt; and for shifting from static text format publications to dynamic knowledge mapping&lt;ref name=&quot;:11&quot;&gt;{{Cite web|url=https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/|title=The death of the literature review and the rise of the dynamic knowledge map - LSE Impact|last=Taster|date=2019-05-14|website=LSE Impact - Understanding impact and practice in academic research|access-date=2026-06-23}}&lt;/ref&gt;. This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : &lt;ref&gt;{{Cite journal|last=Elliott|first=Julian H.|last2=Synnot|first2=Anneliese|last3=Turner|first3=Tari|last4=Simmonds|first4=Mark|last5=Akl|first5=Elie A.|last6=McDonald|first6=Steve|last7=Salanti|first7=Georgia|last8=Meerpohl|first8=Joerg|last9=MacLehose|first9=Harriet|date=2017-11|title=Living systematic review: 1. Introduction—the why, what, when, and how|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435617306364|journal=Journal of Clinical Epidemiology|volume=91|pages=23–30|doi=10.1016/j.jclinepi.2017.08.010|issn=0895-4356}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Uttley|first=Lesley|last2=Quintana|first2=Daniel S.|last3=Montgomery|first3=Paul|last4=Carroll|first4=Christopher|last5=Page|first5=Matthew J.|last6=Falzon|first6=Louise|last7=Sutton|first7=Anthea|last8=Moher|first8=David|date=2023-04|title=The problems with systematic reviews: a living systematic review|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435623000112|journal=Journal of Clinical Epidemiology|volume=156|pages=30–41|doi=10.1016/j.jclinepi.2023.01.011|issn=0895-4356}}&lt;/ref&gt;,&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;). 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.

=== 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&lt;ref&gt;{{Cite journal|last=Haley|first=Usha C. V.|date=2023-09-01|title=Triviality and the Search for Scholarly Impact|url=https://doi.org/10.1177/01708406231175292|journal=Organization Studies|language=EN|volume=44|issue=9|pages=1547–1550|doi=10.1177/01708406231175292|issn=0170-8406}}&lt;/ref&gt;. Wikipedia is a key knowledge dissemination platform widely used by students&lt;ref&gt;{{Cite journal|last=Sunvy|first=Ahmed Shafkat|last2=Reza|first2=Raiyan Bin|date=2023-04-12|title=Students’ Perception of Wikipedia as an Academic Information Source|url=https://ejournal.undiksha.ac.id/index.php/IJERR/article/view/57572|journal=Indonesian Journal Of Educational Research and Review|volume=6|issue=1|pages=134–147|doi=10.23887/ijerr.v6i1.57572|issn=2621-8984}}&lt;/ref&gt; and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature&lt;ref&gt;{{Cite journal|last=Thompson|first=Neil|last2=Hanley|first2=Douglas|date=2017|title=Science Is Shaped by Wikipedia: Evidence from a Randomized Control Trial|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3039505|journal=SSRN Electronic Journal|doi=10.2139/ssrn.3039505|issn=1556-5068}}&lt;/ref&gt; and that some scholars cite directly Wikipedia&lt;ref&gt;{{Cite journal|last=Dooley|first=Patricia L.|date=2010-07-07|title=Wikipedia and the two-faced professoriate|url=https://doi.org/10.1145/1832772.1832803|journal=Proceedings of the 6th International Symposium on Wikis and Open Collaboration|series=WikiSym '10|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=1–2|doi=10.1145/1832772.1832803|isbn=978-1-4503-0056-8}}&lt;/ref&gt;. However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives&lt;ref&gt;{{Cite journal|last=Chen|first=Yan|last2=Farzan|first2=Rosta|last3=Kraut|first3=Robert|last4=YeckehZaare|first4=Iman|last5=Zhang|first5=Ark Fangzhou|date=2024-05|title=Motivating Experts to Contribute to Digital Public Goods: A Personalized Field Experiment on Wikipedia|url=https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4852|journal=Management Science|volume=70|issue=5|pages=3264–3280|doi=10.1287/mnsc.2023.4852|issn=0025-1909}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Kincaid|first=Dustin W.|last2=Beck|first2=Whitney S.|last3=Brandt|first3=Jessica E.|last4=Mars Brisbin|first4=Margaret|last5=Farrell|first5=Kaitlin J.|last6=Hondula|first6=Kelly L.|last7=Larson|first7=Erin I.|last8=Shogren|first8=Arial J.|date=2021|title=Wikipedia can help resolve information inequality in the aquatic sciences|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/lol2.10168|journal=Limnology and Oceanography Letters|language=en|volume=6|issue=1|pages=18–23|doi=10.1002/lol2.10168|issn=2378-2242}}&lt;/ref&gt;, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow&lt;ref name=&quot;:10&quot;&gt;Taraborelli, D., Mietchen, D., Alevizou, P., &amp; Gill, A. (2011, August). Expert participation on Wikipedia: Barriers and opportunities. Wikimania 2011, Haifa, Israel. &lt;nowiki&gt;http://upload.wikimedia.org/wikipedia/commons/4/4f/Expert_Participation_Survey_-_Wikimania_2011.pdf&lt;/nowiki&gt; &lt;/ref&gt;. In addition, expert participation is not immune to the gender gap&lt;ref name=&quot;:10&quot; /&gt;. Because of gender segregation in disciplines&lt;ref&gt;{{Cite journal|last=Ceci|first=Stephen J.|last2=Ginther|first2=Donna K.|last3=Kahn|first3=Shulamit|last4=Williams|first4=Wendy M.|date=2014-12-01|title=Women in Academic Science: A Changing Landscape|url=https://doi.org/10.1177/1529100614541236|journal=Psychological Science in the Public Interest|language=EN|volume=15|issue=3|pages=75–141|doi=10.1177/1529100614541236|issn=1529-1006}}&lt;/ref&gt;, this may be detrimental to the content coverage on “female” topics&lt;ref&gt;{{Cite journal|last=Lam|first=Shyong (Tony) K.|last2=Uduwage|first2=Anuradha|last3=Dong|first3=Zhenhua|last4=Sen|first4=Shilad|last5=Musicant|first5=David R.|last6=Terveen|first6=Loren|last7=Riedl|first7=John|date=2011-10-03|title=WP:clubhouse?: an exploration of Wikipedia's gender imbalance|url=https://dl.acm.org/doi/10.1145/2038558.2038560|language=en|publisher=ACM|pages=1–10|doi=10.1145/2038558.2038560|isbn=978-1-4503-0909-7}}&lt;/ref&gt;, 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 &quot;a fair and equitable process of moving towards a post-carbon society&quot;&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;. 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&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 

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&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:16&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; and fragmented results coming from various disciplines&lt;ref name=&quot;:20&quot;&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;. Then, making literature reviews &quot;living&quot; 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&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, 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&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref name=&quot;:12&quot; /&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but not yet integrated into tested and validated methodologies&lt;ref name=&quot;:13&quot;&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;. Human validation stays notably necessary&lt;ref&gt;{{Cite journal|last=Alshami|first=Ahmad|last2=Elsayed|first2=Moustafa|last3=Ali|first3=Eslam|last4=Eltoukhy|first4=Abdelrahman E. E.|last5=Zayed|first5=Tarek|date=2023-07-09|title=Harnessing the Power of ChatGPT for Automating Systematic Review Process: Methodology, Case Study, Limitations, and Future Directions|url=https://www.mdpi.com/2079-8954/11/7/351|journal=Systems|language=en|volume=11|issue=7|pages=351|doi=10.3390/systems11070351|issn=2079-8954}}&lt;/ref&gt;,&lt;ref name=&quot;:13&quot; /&gt;. 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&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&amp;oldid=1360514388|journal=Wikipedia|language=en}}&lt;/ref&gt;, 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, and a &quot;collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation&quot;.&lt;ref&gt;{{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}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&amp;oldid=1360462340|journal=Wikipedia|language=en}}&lt;/ref&gt; &quot;A [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}&lt;/ref&gt; 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.&quot;&lt;ref&gt;{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}&lt;/ref&gt; Such graphs have a strong potential to conduct knowledge synthesis&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;&lt;ref name=&quot;:18&quot; /&gt;. They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;.  

Wikidata is multilingual... 

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]])&lt;ref&gt;{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&amp;oldid=1358552930|journal=Wikipedia|language=en}}&lt;/ref&gt;. Like all Wikimedia projects, Wikiversity is editable by everyone, have a discussion tab and a history log tab. 

=== Use of Wikidata and Wikiversity for research ===

== 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&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt; (e.g. the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;). 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)&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 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=&quot;wikitable&quot;
|+
!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 &quot;export to Wikidata QuickStatements&quot; 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|&quot;Merge&quot; 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. &lt;ref name=&quot;:5&quot; /&gt;). 

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=&quot;wikitable&quot;
|+
!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}}&lt;ref&gt;{{Cite journal|last=Paré|first=Guy|last2=Trudel|first2=Marie-Claude|last3=Jaana|first3=Mirou|last4=Kitsiou|first4=Spyros|date=2015-03|title=Synthesizing information systems knowledge: A typology of literature reviews|url=https://linkinghub.elsevier.com/retrieve/pii/S0378720614001116|journal=Information &amp; Management|language=en|volume=52|issue=2|pages=183–199|doi=10.1016/j.im.2014.08.008}}&lt;/ref&gt;. After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : 
{| class=&quot;wikitable&quot;
|+
!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 &quot;{{Wikidata entity link|Q137901202}}&quot; 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}}, &quot;energy transition&quot; 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 &quot;filter&quot; 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 :&lt;ref&gt;{{Cite journal|last=Nielsen|first=Finn Årup|last2=Lyhne|first2=Ivar|last3=Garigliotti|first3=Dario|last4=Butzbach|first4=Annika|last5=Ravn Boess|first5=Emilia|last6=Hose|first6=Katja|last7=Kørnøv|first7=Lone|date=2023|title=Environmental impact assessment reports in Wikidata and a Wikibase|url=https://repositum.tuwien.at/handle/20.500.12708/193492|language=en|publisher=CEUR-WS.org|volume=3443|pages=1–8|doi=10.34726/5421}}&lt;/ref&gt;)  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)&lt;ref name=&quot;:19&quot;&gt;Cañas, Alberto J., et al. &quot;CmapTools: A knowledge modeling and sharing environment.&quot; (2004): 125-135. https://thomaseskridge.com/assets/pdf/Canas-2004.pdf&lt;/ref&gt;. Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]&lt;ref name=&quot;:19&quot; /&gt;, Altas.Ti &quot;network&quot; feature...). The &quot;box and arrow&quot; 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”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. 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&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[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.&quot;&lt;ref name=&quot;:21&quot;&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; After identifying themes, qualitative researchers are expected to theorize the &quot;arrows&quot; between themes&lt;ref name=&quot;:21&quot; /&gt;R.esearchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;. 

=== 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 &quot;concept&quot; 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&lt;ref name=&quot;:22&quot; /&gt; 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...&lt;ref name=&quot;:22&quot;&gt;{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}&lt;/ref&gt;, and scholars have advised to take insight from philosophy to work on concepts&lt;ref&gt;{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}&lt;/ref&gt;. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt;. 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 &quot;something&quot; 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 &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, 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&amp;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 &quot;does not have effect&quot; seems contrary to what is listed in &quot;has goal&quot;.) We would however argue this is not a problem because &quot;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&quot;&lt;ref&gt;{{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}}&lt;/ref&gt;. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}&lt;/ref&gt;, as long as they are supported by references&lt;ref&gt;{{Cite journal|last=Amaral|first=Gabriel|last2=Piscopo|first2=Alessandro|last3=Kaffee|first3=Lucie-aimée|last4=Rodrigues|first4=Odinaldo|last5=Simperl|first5=Elena|date=2021-12-31|title=Assessing the Quality of Sources in Wikidata Across Languages: A Hybrid Approach|url=https://dl.acm.org/doi/10.1145/3484828|journal=Journal of Data and Information Quality|language=en|volume=13|issue=4|pages=1–35|doi=10.1145/3484828|issn=1936-1955}}&lt;/ref&gt;.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, the discrepancy between the stated goals of this movement and what it actually achieves is precisely what some authors are critiquing&lt;ref name=&quot;:20&quot; /&gt;.  

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 &quot;management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served&quot;&lt;ref&gt;{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research &amp; Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}&lt;/ref&gt;, &quot;choice&quot; 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 &quot;concept&quot; and &quot;political concept&quot;.]]
===== Concision =====
Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were &quot;in the tree of another value&quot;. 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&lt;nowiki/&gt;) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using &quot;instance of&quot; as transversal property and checking &quot;instance of or subclass of&quot;).

*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 &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;&lt;ref&gt;{{Cite journal|last=Vanegas-Cantarero|first=María M.|date=2020-12-01|title=Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries|url=https://www.wikidata.org/wiki/Q137901196|journal=Energy Research &amp; Social Science|language=English|volume=70|doi=10.1016/J.ERSS.2020.101716}}&lt;/ref&gt;. Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for &quot;final energy consumption&quot;. 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&lt;ref&gt;{{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}}&lt;/ref&gt;. 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 &quot;{{Wikidata entity link|Q127270577}}&quot;). 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 &quot;natures&quot; 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)&lt;ref&gt;Fleetwood, S. (2004). An ontology for organisation and management studies. ''Critical Realist Applications in Organisation and Management Studies'', 27–53.&lt;/ref&gt;. 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 &quot;supports multiple coexisting classification&quot; and allow multiple ontological frameworks to coexist.&lt;ref name=&quot;:8&quot; /&gt; 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=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{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}}&lt;/ref&gt; 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}}.

== 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 statements using a specitic source in order to map the content of a single academic article. First test here: https://w.wiki/PFqH
*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}}).

=== Properties visualisation ===

*Visualise the &quot;tree&quot; 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 &quot;instance of&quot; values for energy democracy in Wikidata.png|center|thumb|553x553px|Visualisation of the &quot;{{Wikidata entity link|P31}}&quot; 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 &quot;topic in context&quot;)[[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 &quot;The topic in context&quot; 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] .]]

=== Mapping sources consensus ===

*Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible.

== 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&lt;ref&gt;{{Cite journal|last=Maryl|first=Maciej|last2=Blaszczyńska|first2=Marta|last3=Zalotyńska|first3=Agnieszka|last4=Taylor|first4=Laurence|last5=Avanço|first5=Karla|last6=Balula|first6=Ana|last7=Buchner|first7=Anna|last8=Caliman|first8=Lorena|last9=Clivaz|first9=Claire|date=2021-01|title=Future of Scholarly Communication|url=https://hal.science/hal-03277615}}&lt;/ref&gt;). 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&lt;ref&gt;{{Cite web|url=https://eprints.whiterose.ac.uk/id/eprint/226771/|title=CLOUDSENSE Collaborative Review of Wikipedia Cloud Feedback article|last=Finney|first=D.|last2=Huang|first2=X.|date=2025-05-16|website=eprints.whiterose.ac.uk|language=en|doi=10.48785/100/329|access-date=2026-07-28|last3=Hinnie|first3=L.|last4=Sheppard|first4=N.}}&lt;/ref&gt;.

=== 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 &quot;remixed&quot; 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&lt;ref&gt;{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}&lt;/ref&gt;. Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to &quot;navigate&quot; 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 &quot;navigate&quot; 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&lt;ref name=&quot;:9&quot; /&gt;. 

=== Quality assessment ===
&lt;ref&gt;{{Citation|vauthors=((E. Peireira, A.))|year=2022|title=Usages et transformations d’un commun numérique : les données généalogiques équines|publisher=Paris 10|url=https://theses.fr/s357002|access-date=25 August 2026}}&lt;/ref&gt;  

=== Future research ===
- Can we &quot;reason&quot; 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 &lt;ref name=&quot;:24&quot;&gt;{{cite journal|date=June 2015|title=Novices’ Struggles with Conceptual and Theoretical Framing in Writing Dissertations and Papers for Publication|journal=Publications|publisher=Multidisciplinary Digital Publishing Institute|volume=3|issue=2|pages=104–119|doi=10.3390/publications3020104|issn=2304-6775|vauthors=((Casanave, C. P.)), ((Li, Y.))}}&lt;/ref&gt;. Conceptualization require to answer the question « what are our data examples of? » &lt;ref name=&quot;:24&quot; /&gt;.  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 &lt;ref name=&quot;:24&quot; /&gt;. 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_&amp;_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=&quot;wikitable sortable&quot;
! 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}}</text>
      <sha1>0nq5n3wkqxgn9eo6mbtxdg9z16j3elj</sha1>
    </revision>
    <revision>
      <id>2829771</id>
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      <timestamp>2026-08-30T20:31:35Z</timestamp>
      <contributor>
        <username>Jeanne Noiraud</username>
        <id>1366702</id>
      </contributor>
      <comment>/* Knowledge modelling in Wikidata : second round of analysis */ explaining methodology step</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="114902" sha1="oofi9szbv50l33k843naguoyrkp7hph" xml:space="preserve">== Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) ==

== Acknowledgements ==
The present text was originally written on a Wikiversity page, if you are reading it in another format, you can find this page here : [[Just sustainability transitions: a living review|https://en.wikiversity.org/wiki/Just_sustainability_transitions:_a_living_review]]. You are free to add your comments on the page discussion section. 

=== Contributors ===
{| class=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-3-030-89460-3_2|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021|publisher=Springer International Publishing|isbn=978-3-030-89460-3|editor-last=Heffron|editor-first=Raphael J.|location=Cham|pages=9–19|language=en|doi=10.1007/978-3-030-89460-3_2}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.sciencedirect.com/science/article/pii/S0301421518302301|journal=Energy Policy|volume=119|pages=1–7|doi=10.1016/j.enpol.2018.04.014|issn=0301-4215}}&lt;/ref&gt;. 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&lt;ref name=&quot;:15&quot;&gt;{{Cite journal|date=2020-03-01|title=Micro-foundations of the multi-level perspective on socio-technical transitions: Developing a multi-dimensional model of agency through crossovers between social constructivism, evolutionary economics and neo-institutional theory|url=https://www.sciencedirect.com/science/article/abs/pii/S0040162518316111|journal=Technological Forecasting and Social Change|language=en-US|volume=152|pages=119894|doi=10.1016/j.techfore.2019.119894|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2023-08-01|title=A socio-technical transition perspective on positive tipping points in climate change mitigation: Analysing seven interacting feedback loops in offshore wind and electric vehicles acceleration|url=https://www.sciencedirect.com/science/article/pii/S0040162523003244|journal=Technological Forecasting and Social Change|language=en-US|volume=193|pages=122639|doi=10.1016/j.techfore.2023.122639|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Sovacool|first=Benjamin K.|last2=Geels|first2=Frank W.|last3=Andersen|first3=Allan Dahl|last4=Grubb|first4=Michael|last5=Jordan|first5=Andrew J.|last6=Kern|first6=Florian|last7=Kivimaa|first7=Paula|last8=Lockwood|first8=Matthew|last9=Markard|first9=Jochen|date=2025-03-01|title=The acceleration of low-carbon transitions: Insights, concepts, challenges, and new directions for research|url=https://www.sciencedirect.com/science/article/pii/S2214629625000295|journal=Energy Research &amp; Social Science|volume=121|pages=103948|doi=10.1016/j.erss.2025.103948|issn=2214-6296}}&lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Kepes|first=Sven|last2=Bennett|first2=Andrew A.|last3=McDaniel|first3=Michael A.|date=2014-09|title=Evidence-Based Management and the Trustworthiness of Our Cumulative Scientific Knowledge: Implications for Teaching, Research, and Practice|url=https://journals.aom.org/doi/10.5465/amle.2013.0193|journal=Academy of Management Learning &amp; Education|volume=13|issue=3|pages=446–466|doi=10.5465/amle.2013.0193|issn=1537-260X}}&lt;/ref&gt;&lt;ref&gt;Pfeffer, J., &amp; Sutton, R. I. (2006). Evidence-Based Management. Harvard Business Review, 13. &lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Bornmann|first=Lutz|last2=Mutz|first2=Rüdiger|date=2015|title=Growth rates of modern science: A bibliometric analysis based on the number of publications and cited references|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/asi.23329|journal=Journal of the Association for Information Science and Technology|language=en|volume=66|issue=11|pages=2215–2222|doi=10.1002/asi.23329|issn=2330-1643}}&lt;/ref&gt;&lt;ref name=&quot;:23&quot;&gt;{{Cite journal|last=Hanson|first=Mark A.|last2=Barreiro|first2=Pablo Gómez|last3=Crosetto|first3=Paolo|last4=Brockington|first4=Dan|date=2024-11-01|title=The strain on scientific publishing|url=https://doi.org/10.1162/qss_a_00327|journal=Quantitative Science Studies|volume=5|issue=4|pages=823–843|doi=10.1162/qss_a_00327|issn=2641-3337}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=http://blogs.nature.com/news/2014/05/global-scientific-output-doubles-every-nine-years.html|title=Global scientific output doubles every nine years : News blog|website=blogs.nature.com|language=en-US|access-date=2026-06-23}}&lt;/ref&gt;, 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&lt;ref name=&quot;:23&quot; /&gt;. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge&lt;ref&gt;{{Cite journal|last=Hartman|first=Yvonne|last2=Darab|first2=Sandy|date=2012-01-01|title=A Call for Slow Scholarship: A Case Study on the Intensification of Academic Life and Its Implications for Pedagogy|url=https://doi.org/10.1080/10714413.2012.643740|journal=Review of Education, Pedagogy, and Cultural Studies|volume=34|issue=1-2|pages=49–60|doi=10.1080/10714413.2012.643740|issn=1071-4413}}&lt;/ref&gt;. “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&lt;ref&gt;{{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)}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=L.|first=Mobley, David|last2=M.|first2=Zuckerman, Daniel|date=2015-02-03|title=A proposal for regularly updated review/survey articles: &quot;Perpetual Reviews&quot;|url=https://arxiv.org/abs/1502.01329|journal=arXiv.org|language=en|doi=10.48550/arXiv.1502.01329}}&lt;/ref&gt; and for shifting from static text format publications to dynamic knowledge mapping&lt;ref name=&quot;:11&quot;&gt;{{Cite web|url=https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/|title=The death of the literature review and the rise of the dynamic knowledge map - LSE Impact|last=Taster|date=2019-05-14|website=LSE Impact - Understanding impact and practice in academic research|access-date=2026-06-23}}&lt;/ref&gt;. This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : &lt;ref&gt;{{Cite journal|last=Elliott|first=Julian H.|last2=Synnot|first2=Anneliese|last3=Turner|first3=Tari|last4=Simmonds|first4=Mark|last5=Akl|first5=Elie A.|last6=McDonald|first6=Steve|last7=Salanti|first7=Georgia|last8=Meerpohl|first8=Joerg|last9=MacLehose|first9=Harriet|date=2017-11|title=Living systematic review: 1. Introduction—the why, what, when, and how|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435617306364|journal=Journal of Clinical Epidemiology|volume=91|pages=23–30|doi=10.1016/j.jclinepi.2017.08.010|issn=0895-4356}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Uttley|first=Lesley|last2=Quintana|first2=Daniel S.|last3=Montgomery|first3=Paul|last4=Carroll|first4=Christopher|last5=Page|first5=Matthew J.|last6=Falzon|first6=Louise|last7=Sutton|first7=Anthea|last8=Moher|first8=David|date=2023-04|title=The problems with systematic reviews: a living systematic review|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435623000112|journal=Journal of Clinical Epidemiology|volume=156|pages=30–41|doi=10.1016/j.jclinepi.2023.01.011|issn=0895-4356}}&lt;/ref&gt;,&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;). 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.

=== 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&lt;ref&gt;{{Cite journal|last=Haley|first=Usha C. V.|date=2023-09-01|title=Triviality and the Search for Scholarly Impact|url=https://doi.org/10.1177/01708406231175292|journal=Organization Studies|language=EN|volume=44|issue=9|pages=1547–1550|doi=10.1177/01708406231175292|issn=0170-8406}}&lt;/ref&gt;. Wikipedia is a key knowledge dissemination platform widely used by students&lt;ref&gt;{{Cite journal|last=Sunvy|first=Ahmed Shafkat|last2=Reza|first2=Raiyan Bin|date=2023-04-12|title=Students’ Perception of Wikipedia as an Academic Information Source|url=https://ejournal.undiksha.ac.id/index.php/IJERR/article/view/57572|journal=Indonesian Journal Of Educational Research and Review|volume=6|issue=1|pages=134–147|doi=10.23887/ijerr.v6i1.57572|issn=2621-8984}}&lt;/ref&gt; and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature&lt;ref&gt;{{Cite journal|last=Thompson|first=Neil|last2=Hanley|first2=Douglas|date=2017|title=Science Is Shaped by Wikipedia: Evidence from a Randomized Control Trial|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3039505|journal=SSRN Electronic Journal|doi=10.2139/ssrn.3039505|issn=1556-5068}}&lt;/ref&gt; and that some scholars cite directly Wikipedia&lt;ref&gt;{{Cite journal|last=Dooley|first=Patricia L.|date=2010-07-07|title=Wikipedia and the two-faced professoriate|url=https://doi.org/10.1145/1832772.1832803|journal=Proceedings of the 6th International Symposium on Wikis and Open Collaboration|series=WikiSym '10|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=1–2|doi=10.1145/1832772.1832803|isbn=978-1-4503-0056-8}}&lt;/ref&gt;. However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives&lt;ref&gt;{{Cite journal|last=Chen|first=Yan|last2=Farzan|first2=Rosta|last3=Kraut|first3=Robert|last4=YeckehZaare|first4=Iman|last5=Zhang|first5=Ark Fangzhou|date=2024-05|title=Motivating Experts to Contribute to Digital Public Goods: A Personalized Field Experiment on Wikipedia|url=https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4852|journal=Management Science|volume=70|issue=5|pages=3264–3280|doi=10.1287/mnsc.2023.4852|issn=0025-1909}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Kincaid|first=Dustin W.|last2=Beck|first2=Whitney S.|last3=Brandt|first3=Jessica E.|last4=Mars Brisbin|first4=Margaret|last5=Farrell|first5=Kaitlin J.|last6=Hondula|first6=Kelly L.|last7=Larson|first7=Erin I.|last8=Shogren|first8=Arial J.|date=2021|title=Wikipedia can help resolve information inequality in the aquatic sciences|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/lol2.10168|journal=Limnology and Oceanography Letters|language=en|volume=6|issue=1|pages=18–23|doi=10.1002/lol2.10168|issn=2378-2242}}&lt;/ref&gt;, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow&lt;ref name=&quot;:10&quot;&gt;Taraborelli, D., Mietchen, D., Alevizou, P., &amp; Gill, A. (2011, August). Expert participation on Wikipedia: Barriers and opportunities. Wikimania 2011, Haifa, Israel. &lt;nowiki&gt;http://upload.wikimedia.org/wikipedia/commons/4/4f/Expert_Participation_Survey_-_Wikimania_2011.pdf&lt;/nowiki&gt; &lt;/ref&gt;. In addition, expert participation is not immune to the gender gap&lt;ref name=&quot;:10&quot; /&gt;. Because of gender segregation in disciplines&lt;ref&gt;{{Cite journal|last=Ceci|first=Stephen J.|last2=Ginther|first2=Donna K.|last3=Kahn|first3=Shulamit|last4=Williams|first4=Wendy M.|date=2014-12-01|title=Women in Academic Science: A Changing Landscape|url=https://doi.org/10.1177/1529100614541236|journal=Psychological Science in the Public Interest|language=EN|volume=15|issue=3|pages=75–141|doi=10.1177/1529100614541236|issn=1529-1006}}&lt;/ref&gt;, this may be detrimental to the content coverage on “female” topics&lt;ref&gt;{{Cite journal|last=Lam|first=Shyong (Tony) K.|last2=Uduwage|first2=Anuradha|last3=Dong|first3=Zhenhua|last4=Sen|first4=Shilad|last5=Musicant|first5=David R.|last6=Terveen|first6=Loren|last7=Riedl|first7=John|date=2011-10-03|title=WP:clubhouse?: an exploration of Wikipedia's gender imbalance|url=https://dl.acm.org/doi/10.1145/2038558.2038560|language=en|publisher=ACM|pages=1–10|doi=10.1145/2038558.2038560|isbn=978-1-4503-0909-7}}&lt;/ref&gt;, 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 &quot;a fair and equitable process of moving towards a post-carbon society&quot;&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;. 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&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 

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&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:16&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; and fragmented results coming from various disciplines&lt;ref name=&quot;:20&quot;&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;. Then, making literature reviews &quot;living&quot; 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&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, 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&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref name=&quot;:12&quot; /&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but not yet integrated into tested and validated methodologies&lt;ref name=&quot;:13&quot;&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;. Human validation stays notably necessary&lt;ref&gt;{{Cite journal|last=Alshami|first=Ahmad|last2=Elsayed|first2=Moustafa|last3=Ali|first3=Eslam|last4=Eltoukhy|first4=Abdelrahman E. E.|last5=Zayed|first5=Tarek|date=2023-07-09|title=Harnessing the Power of ChatGPT for Automating Systematic Review Process: Methodology, Case Study, Limitations, and Future Directions|url=https://www.mdpi.com/2079-8954/11/7/351|journal=Systems|language=en|volume=11|issue=7|pages=351|doi=10.3390/systems11070351|issn=2079-8954}}&lt;/ref&gt;,&lt;ref name=&quot;:13&quot; /&gt;. 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&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&amp;oldid=1360514388|journal=Wikipedia|language=en}}&lt;/ref&gt;, 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, and a &quot;collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation&quot;.&lt;ref&gt;{{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}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&amp;oldid=1360462340|journal=Wikipedia|language=en}}&lt;/ref&gt; &quot;A [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}&lt;/ref&gt; 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.&quot;&lt;ref&gt;{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}&lt;/ref&gt; Such graphs have a strong potential to conduct knowledge synthesis&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;&lt;ref name=&quot;:18&quot; /&gt;. They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;.  

Wikidata is multilingual... 

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]])&lt;ref&gt;{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&amp;oldid=1358552930|journal=Wikipedia|language=en}}&lt;/ref&gt;. Like all Wikimedia projects, Wikiversity is editable by everyone, have a discussion tab and a history log tab. 

=== Use of Wikidata and Wikiversity for research ===

== 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&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt; (e.g. the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;). 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)&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 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=&quot;wikitable&quot;
|+
!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 &quot;export to Wikidata QuickStatements&quot; 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|&quot;Merge&quot; 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. &lt;ref name=&quot;:5&quot; /&gt;). 

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=&quot;wikitable&quot;
|+
!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}}&lt;ref&gt;{{Cite journal|last=Paré|first=Guy|last2=Trudel|first2=Marie-Claude|last3=Jaana|first3=Mirou|last4=Kitsiou|first4=Spyros|date=2015-03|title=Synthesizing information systems knowledge: A typology of literature reviews|url=https://linkinghub.elsevier.com/retrieve/pii/S0378720614001116|journal=Information &amp; Management|language=en|volume=52|issue=2|pages=183–199|doi=10.1016/j.im.2014.08.008}}&lt;/ref&gt;. After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : 
{| class=&quot;wikitable&quot;
|+
!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 &quot;{{Wikidata entity link|Q137901202}}&quot; 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}}, &quot;energy transition&quot; 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 &quot;filter&quot; 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 :&lt;ref&gt;{{Cite journal|last=Nielsen|first=Finn Årup|last2=Lyhne|first2=Ivar|last3=Garigliotti|first3=Dario|last4=Butzbach|first4=Annika|last5=Ravn Boess|first5=Emilia|last6=Hose|first6=Katja|last7=Kørnøv|first7=Lone|date=2023|title=Environmental impact assessment reports in Wikidata and a Wikibase|url=https://repositum.tuwien.at/handle/20.500.12708/193492|language=en|publisher=CEUR-WS.org|volume=3443|pages=1–8|doi=10.34726/5421}}&lt;/ref&gt;)  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)&lt;ref name=&quot;:19&quot;&gt;Cañas, Alberto J., et al. &quot;CmapTools: A knowledge modeling and sharing environment.&quot; (2004): 125-135. https://thomaseskridge.com/assets/pdf/Canas-2004.pdf&lt;/ref&gt;. Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]&lt;ref name=&quot;:19&quot; /&gt;, Altas.Ti &quot;network&quot; feature...). The &quot;box and arrow&quot; 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”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. 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&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[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.&quot;&lt;ref name=&quot;:21&quot;&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; After identifying themes, qualitative researchers are expected to theorize the &quot;arrows&quot; between themes&lt;ref name=&quot;:21&quot; /&gt;R.esearchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;. 

=== 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 &quot;concept&quot; 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&lt;ref name=&quot;:22&quot; /&gt; 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...&lt;ref name=&quot;:22&quot;&gt;{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}&lt;/ref&gt;, and scholars have advised to take insight from philosophy to work on concepts&lt;ref&gt;{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}&lt;/ref&gt;. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt;. 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 &quot;something&quot; 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 &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, 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&amp;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 &quot;does not have effect&quot; seems contrary to what is listed in &quot;has goal&quot;.) We would however argue this is not a problem because &quot;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&quot;&lt;ref&gt;{{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}}&lt;/ref&gt;. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}&lt;/ref&gt;, as long as they are supported by references&lt;ref&gt;{{Cite journal|last=Amaral|first=Gabriel|last2=Piscopo|first2=Alessandro|last3=Kaffee|first3=Lucie-aimée|last4=Rodrigues|first4=Odinaldo|last5=Simperl|first5=Elena|date=2021-12-31|title=Assessing the Quality of Sources in Wikidata Across Languages: A Hybrid Approach|url=https://dl.acm.org/doi/10.1145/3484828|journal=Journal of Data and Information Quality|language=en|volume=13|issue=4|pages=1–35|doi=10.1145/3484828|issn=1936-1955}}&lt;/ref&gt;.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, the discrepancy between the stated goals of this movement and what it actually achieves is precisely what some authors are critiquing&lt;ref name=&quot;:20&quot; /&gt;.  

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 &quot;management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served&quot;&lt;ref&gt;{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research &amp; Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}&lt;/ref&gt;, &quot;choice&quot; 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 &quot;concept&quot; and &quot;political concept&quot;.]]
===== Concision =====
Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were &quot;in the tree of another value&quot;. 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&lt;nowiki/&gt;) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using &quot;instance of&quot; as transversal property and checking &quot;instance of or subclass of&quot;).

*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 &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;&lt;ref&gt;{{Cite journal|last=Vanegas-Cantarero|first=María M.|date=2020-12-01|title=Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries|url=https://www.wikidata.org/wiki/Q137901196|journal=Energy Research &amp; Social Science|language=English|volume=70|doi=10.1016/J.ERSS.2020.101716}}&lt;/ref&gt;. Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for &quot;final energy consumption&quot;. 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&lt;ref&gt;{{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}}&lt;/ref&gt;. 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 &quot;{{Wikidata entity link|Q127270577}}&quot;). 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 &quot;natures&quot; 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)&lt;ref&gt;Fleetwood, S. (2004). An ontology for organisation and management studies. ''Critical Realist Applications in Organisation and Management Studies'', 27–53.&lt;/ref&gt;. 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 &quot;supports multiple coexisting classification&quot; and allow multiple ontological frameworks to coexist.&lt;ref name=&quot;:8&quot; /&gt; 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=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{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}}&lt;/ref&gt; 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 focused on identifying key concepts, and creating them when necessary (ex: {{Wikidata entity link|Q141223829}}).

== 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 statements using a specitic source in order to map the content of a single academic article. First test here: https://w.wiki/PFqH
*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}}).

=== Properties visualisation ===

*Visualise the &quot;tree&quot; 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 &quot;instance of&quot; values for energy democracy in Wikidata.png|center|thumb|553x553px|Visualisation of the &quot;{{Wikidata entity link|P31}}&quot; 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 &quot;topic in context&quot;)[[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 &quot;The topic in context&quot; 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] .]]

=== Mapping sources consensus ===

*Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible.

== 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&lt;ref&gt;{{Cite journal|last=Maryl|first=Maciej|last2=Blaszczyńska|first2=Marta|last3=Zalotyńska|first3=Agnieszka|last4=Taylor|first4=Laurence|last5=Avanço|first5=Karla|last6=Balula|first6=Ana|last7=Buchner|first7=Anna|last8=Caliman|first8=Lorena|last9=Clivaz|first9=Claire|date=2021-01|title=Future of Scholarly Communication|url=https://hal.science/hal-03277615}}&lt;/ref&gt;). 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&lt;ref&gt;{{Cite web|url=https://eprints.whiterose.ac.uk/id/eprint/226771/|title=CLOUDSENSE Collaborative Review of Wikipedia Cloud Feedback article|last=Finney|first=D.|last2=Huang|first2=X.|date=2025-05-16|website=eprints.whiterose.ac.uk|language=en|doi=10.48785/100/329|access-date=2026-07-28|last3=Hinnie|first3=L.|last4=Sheppard|first4=N.}}&lt;/ref&gt;.

=== 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 &quot;remixed&quot; 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&lt;ref&gt;{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}&lt;/ref&gt;. Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to &quot;navigate&quot; 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 &quot;navigate&quot; 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&lt;ref name=&quot;:9&quot; /&gt;. 

=== Quality assessment ===
&lt;ref&gt;{{Citation|vauthors=((E. Peireira, A.))|year=2022|title=Usages et transformations d’un commun numérique : les données généalogiques équines|publisher=Paris 10|url=https://theses.fr/s357002|access-date=25 August 2026}}&lt;/ref&gt;  

=== Future research ===
- Can we &quot;reason&quot; 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 &lt;ref name=&quot;:24&quot;&gt;{{cite journal|date=June 2015|title=Novices’ Struggles with Conceptual and Theoretical Framing in Writing Dissertations and Papers for Publication|journal=Publications|publisher=Multidisciplinary Digital Publishing Institute|volume=3|issue=2|pages=104–119|doi=10.3390/publications3020104|issn=2304-6775|vauthors=((Casanave, C. P.)), ((Li, Y.))}}&lt;/ref&gt;. Conceptualization require to answer the question « what are our data examples of? » &lt;ref name=&quot;:24&quot; /&gt;.  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 &lt;ref name=&quot;:24&quot; /&gt;. 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_&amp;_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=&quot;wikitable sortable&quot;
! 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
|-
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|}


== References ==
{{References}}</text>
      <sha1>oofi9szbv50l33k843naguoyrkp7hph</sha1>
    </revision>
    <revision>
      <id>2829781</id>
      <parentid>2829771</parentid>
      <timestamp>2026-08-30T22:03:31Z</timestamp>
      <contributor>
        <username>Jeanne Noiraud</username>
        <id>1366702</id>
      </contributor>
      <comment>/* Knowledge modelling in Wikidata : second round of analysis */ describing research steps</comment>
      <origin>2829781</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="115110" sha1="h29tbo1371j59rnxhbhtmxgyl1eu0rr" xml:space="preserve">== Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) ==

== Acknowledgements ==
The present text was originally written on a Wikiversity page, if you are reading it in another format, you can find this page here : [[Just sustainability transitions: a living review|https://en.wikiversity.org/wiki/Just_sustainability_transitions:_a_living_review]]. You are free to add your comments on the page discussion section. 

=== Contributors ===
{| class=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-3-030-89460-3_2|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021|publisher=Springer International Publishing|isbn=978-3-030-89460-3|editor-last=Heffron|editor-first=Raphael J.|location=Cham|pages=9–19|language=en|doi=10.1007/978-3-030-89460-3_2}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.sciencedirect.com/science/article/pii/S0301421518302301|journal=Energy Policy|volume=119|pages=1–7|doi=10.1016/j.enpol.2018.04.014|issn=0301-4215}}&lt;/ref&gt;. 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&lt;ref name=&quot;:15&quot;&gt;{{Cite journal|date=2020-03-01|title=Micro-foundations of the multi-level perspective on socio-technical transitions: Developing a multi-dimensional model of agency through crossovers between social constructivism, evolutionary economics and neo-institutional theory|url=https://www.sciencedirect.com/science/article/abs/pii/S0040162518316111|journal=Technological Forecasting and Social Change|language=en-US|volume=152|pages=119894|doi=10.1016/j.techfore.2019.119894|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2023-08-01|title=A socio-technical transition perspective on positive tipping points in climate change mitigation: Analysing seven interacting feedback loops in offshore wind and electric vehicles acceleration|url=https://www.sciencedirect.com/science/article/pii/S0040162523003244|journal=Technological Forecasting and Social Change|language=en-US|volume=193|pages=122639|doi=10.1016/j.techfore.2023.122639|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Sovacool|first=Benjamin K.|last2=Geels|first2=Frank W.|last3=Andersen|first3=Allan Dahl|last4=Grubb|first4=Michael|last5=Jordan|first5=Andrew J.|last6=Kern|first6=Florian|last7=Kivimaa|first7=Paula|last8=Lockwood|first8=Matthew|last9=Markard|first9=Jochen|date=2025-03-01|title=The acceleration of low-carbon transitions: Insights, concepts, challenges, and new directions for research|url=https://www.sciencedirect.com/science/article/pii/S2214629625000295|journal=Energy Research &amp; Social Science|volume=121|pages=103948|doi=10.1016/j.erss.2025.103948|issn=2214-6296}}&lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Kepes|first=Sven|last2=Bennett|first2=Andrew A.|last3=McDaniel|first3=Michael A.|date=2014-09|title=Evidence-Based Management and the Trustworthiness of Our Cumulative Scientific Knowledge: Implications for Teaching, Research, and Practice|url=https://journals.aom.org/doi/10.5465/amle.2013.0193|journal=Academy of Management Learning &amp; Education|volume=13|issue=3|pages=446–466|doi=10.5465/amle.2013.0193|issn=1537-260X}}&lt;/ref&gt;&lt;ref&gt;Pfeffer, J., &amp; Sutton, R. I. (2006). Evidence-Based Management. Harvard Business Review, 13. &lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Bornmann|first=Lutz|last2=Mutz|first2=Rüdiger|date=2015|title=Growth rates of modern science: A bibliometric analysis based on the number of publications and cited references|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/asi.23329|journal=Journal of the Association for Information Science and Technology|language=en|volume=66|issue=11|pages=2215–2222|doi=10.1002/asi.23329|issn=2330-1643}}&lt;/ref&gt;&lt;ref name=&quot;:23&quot;&gt;{{Cite journal|last=Hanson|first=Mark A.|last2=Barreiro|first2=Pablo Gómez|last3=Crosetto|first3=Paolo|last4=Brockington|first4=Dan|date=2024-11-01|title=The strain on scientific publishing|url=https://doi.org/10.1162/qss_a_00327|journal=Quantitative Science Studies|volume=5|issue=4|pages=823–843|doi=10.1162/qss_a_00327|issn=2641-3337}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=http://blogs.nature.com/news/2014/05/global-scientific-output-doubles-every-nine-years.html|title=Global scientific output doubles every nine years : News blog|website=blogs.nature.com|language=en-US|access-date=2026-06-23}}&lt;/ref&gt;, 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&lt;ref name=&quot;:23&quot; /&gt;. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge&lt;ref&gt;{{Cite journal|last=Hartman|first=Yvonne|last2=Darab|first2=Sandy|date=2012-01-01|title=A Call for Slow Scholarship: A Case Study on the Intensification of Academic Life and Its Implications for Pedagogy|url=https://doi.org/10.1080/10714413.2012.643740|journal=Review of Education, Pedagogy, and Cultural Studies|volume=34|issue=1-2|pages=49–60|doi=10.1080/10714413.2012.643740|issn=1071-4413}}&lt;/ref&gt;. “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&lt;ref&gt;{{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)}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=L.|first=Mobley, David|last2=M.|first2=Zuckerman, Daniel|date=2015-02-03|title=A proposal for regularly updated review/survey articles: &quot;Perpetual Reviews&quot;|url=https://arxiv.org/abs/1502.01329|journal=arXiv.org|language=en|doi=10.48550/arXiv.1502.01329}}&lt;/ref&gt; and for shifting from static text format publications to dynamic knowledge mapping&lt;ref name=&quot;:11&quot;&gt;{{Cite web|url=https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/|title=The death of the literature review and the rise of the dynamic knowledge map - LSE Impact|last=Taster|date=2019-05-14|website=LSE Impact - Understanding impact and practice in academic research|access-date=2026-06-23}}&lt;/ref&gt;. This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : &lt;ref&gt;{{Cite journal|last=Elliott|first=Julian H.|last2=Synnot|first2=Anneliese|last3=Turner|first3=Tari|last4=Simmonds|first4=Mark|last5=Akl|first5=Elie A.|last6=McDonald|first6=Steve|last7=Salanti|first7=Georgia|last8=Meerpohl|first8=Joerg|last9=MacLehose|first9=Harriet|date=2017-11|title=Living systematic review: 1. Introduction—the why, what, when, and how|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435617306364|journal=Journal of Clinical Epidemiology|volume=91|pages=23–30|doi=10.1016/j.jclinepi.2017.08.010|issn=0895-4356}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Uttley|first=Lesley|last2=Quintana|first2=Daniel S.|last3=Montgomery|first3=Paul|last4=Carroll|first4=Christopher|last5=Page|first5=Matthew J.|last6=Falzon|first6=Louise|last7=Sutton|first7=Anthea|last8=Moher|first8=David|date=2023-04|title=The problems with systematic reviews: a living systematic review|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435623000112|journal=Journal of Clinical Epidemiology|volume=156|pages=30–41|doi=10.1016/j.jclinepi.2023.01.011|issn=0895-4356}}&lt;/ref&gt;,&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;). 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.

=== 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&lt;ref&gt;{{Cite journal|last=Haley|first=Usha C. V.|date=2023-09-01|title=Triviality and the Search for Scholarly Impact|url=https://doi.org/10.1177/01708406231175292|journal=Organization Studies|language=EN|volume=44|issue=9|pages=1547–1550|doi=10.1177/01708406231175292|issn=0170-8406}}&lt;/ref&gt;. Wikipedia is a key knowledge dissemination platform widely used by students&lt;ref&gt;{{Cite journal|last=Sunvy|first=Ahmed Shafkat|last2=Reza|first2=Raiyan Bin|date=2023-04-12|title=Students’ Perception of Wikipedia as an Academic Information Source|url=https://ejournal.undiksha.ac.id/index.php/IJERR/article/view/57572|journal=Indonesian Journal Of Educational Research and Review|volume=6|issue=1|pages=134–147|doi=10.23887/ijerr.v6i1.57572|issn=2621-8984}}&lt;/ref&gt; and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature&lt;ref&gt;{{Cite journal|last=Thompson|first=Neil|last2=Hanley|first2=Douglas|date=2017|title=Science Is Shaped by Wikipedia: Evidence from a Randomized Control Trial|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3039505|journal=SSRN Electronic Journal|doi=10.2139/ssrn.3039505|issn=1556-5068}}&lt;/ref&gt; and that some scholars cite directly Wikipedia&lt;ref&gt;{{Cite journal|last=Dooley|first=Patricia L.|date=2010-07-07|title=Wikipedia and the two-faced professoriate|url=https://doi.org/10.1145/1832772.1832803|journal=Proceedings of the 6th International Symposium on Wikis and Open Collaboration|series=WikiSym '10|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=1–2|doi=10.1145/1832772.1832803|isbn=978-1-4503-0056-8}}&lt;/ref&gt;. However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives&lt;ref&gt;{{Cite journal|last=Chen|first=Yan|last2=Farzan|first2=Rosta|last3=Kraut|first3=Robert|last4=YeckehZaare|first4=Iman|last5=Zhang|first5=Ark Fangzhou|date=2024-05|title=Motivating Experts to Contribute to Digital Public Goods: A Personalized Field Experiment on Wikipedia|url=https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4852|journal=Management Science|volume=70|issue=5|pages=3264–3280|doi=10.1287/mnsc.2023.4852|issn=0025-1909}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Kincaid|first=Dustin W.|last2=Beck|first2=Whitney S.|last3=Brandt|first3=Jessica E.|last4=Mars Brisbin|first4=Margaret|last5=Farrell|first5=Kaitlin J.|last6=Hondula|first6=Kelly L.|last7=Larson|first7=Erin I.|last8=Shogren|first8=Arial J.|date=2021|title=Wikipedia can help resolve information inequality in the aquatic sciences|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/lol2.10168|journal=Limnology and Oceanography Letters|language=en|volume=6|issue=1|pages=18–23|doi=10.1002/lol2.10168|issn=2378-2242}}&lt;/ref&gt;, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow&lt;ref name=&quot;:10&quot;&gt;Taraborelli, D., Mietchen, D., Alevizou, P., &amp; Gill, A. (2011, August). Expert participation on Wikipedia: Barriers and opportunities. Wikimania 2011, Haifa, Israel. &lt;nowiki&gt;http://upload.wikimedia.org/wikipedia/commons/4/4f/Expert_Participation_Survey_-_Wikimania_2011.pdf&lt;/nowiki&gt; &lt;/ref&gt;. In addition, expert participation is not immune to the gender gap&lt;ref name=&quot;:10&quot; /&gt;. Because of gender segregation in disciplines&lt;ref&gt;{{Cite journal|last=Ceci|first=Stephen J.|last2=Ginther|first2=Donna K.|last3=Kahn|first3=Shulamit|last4=Williams|first4=Wendy M.|date=2014-12-01|title=Women in Academic Science: A Changing Landscape|url=https://doi.org/10.1177/1529100614541236|journal=Psychological Science in the Public Interest|language=EN|volume=15|issue=3|pages=75–141|doi=10.1177/1529100614541236|issn=1529-1006}}&lt;/ref&gt;, this may be detrimental to the content coverage on “female” topics&lt;ref&gt;{{Cite journal|last=Lam|first=Shyong (Tony) K.|last2=Uduwage|first2=Anuradha|last3=Dong|first3=Zhenhua|last4=Sen|first4=Shilad|last5=Musicant|first5=David R.|last6=Terveen|first6=Loren|last7=Riedl|first7=John|date=2011-10-03|title=WP:clubhouse?: an exploration of Wikipedia's gender imbalance|url=https://dl.acm.org/doi/10.1145/2038558.2038560|language=en|publisher=ACM|pages=1–10|doi=10.1145/2038558.2038560|isbn=978-1-4503-0909-7}}&lt;/ref&gt;, 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 &quot;a fair and equitable process of moving towards a post-carbon society&quot;&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;. 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&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 

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&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:16&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; and fragmented results coming from various disciplines&lt;ref name=&quot;:20&quot;&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;. Then, making literature reviews &quot;living&quot; 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&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, 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&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref name=&quot;:12&quot; /&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but not yet integrated into tested and validated methodologies&lt;ref name=&quot;:13&quot;&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;. Human validation stays notably necessary&lt;ref&gt;{{Cite journal|last=Alshami|first=Ahmad|last2=Elsayed|first2=Moustafa|last3=Ali|first3=Eslam|last4=Eltoukhy|first4=Abdelrahman E. E.|last5=Zayed|first5=Tarek|date=2023-07-09|title=Harnessing the Power of ChatGPT for Automating Systematic Review Process: Methodology, Case Study, Limitations, and Future Directions|url=https://www.mdpi.com/2079-8954/11/7/351|journal=Systems|language=en|volume=11|issue=7|pages=351|doi=10.3390/systems11070351|issn=2079-8954}}&lt;/ref&gt;,&lt;ref name=&quot;:13&quot; /&gt;. 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&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&amp;oldid=1360514388|journal=Wikipedia|language=en}}&lt;/ref&gt;, 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, and a &quot;collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation&quot;.&lt;ref&gt;{{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}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&amp;oldid=1360462340|journal=Wikipedia|language=en}}&lt;/ref&gt; &quot;A [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}&lt;/ref&gt; 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.&quot;&lt;ref&gt;{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}&lt;/ref&gt; Such graphs have a strong potential to conduct knowledge synthesis&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;&lt;ref name=&quot;:18&quot; /&gt;. They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;.  

Wikidata is multilingual... 

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]])&lt;ref&gt;{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&amp;oldid=1358552930|journal=Wikipedia|language=en}}&lt;/ref&gt;. Like all Wikimedia projects, Wikiversity is editable by everyone, have a discussion tab and a history log tab. 

=== Use of Wikidata and Wikiversity for research ===

== 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&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt; (e.g. the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;). 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)&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 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=&quot;wikitable&quot;
|+
!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 &quot;export to Wikidata QuickStatements&quot; 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|&quot;Merge&quot; 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. &lt;ref name=&quot;:5&quot; /&gt;). 

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=&quot;wikitable&quot;
|+
!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}}&lt;ref&gt;{{Cite journal|last=Paré|first=Guy|last2=Trudel|first2=Marie-Claude|last3=Jaana|first3=Mirou|last4=Kitsiou|first4=Spyros|date=2015-03|title=Synthesizing information systems knowledge: A typology of literature reviews|url=https://linkinghub.elsevier.com/retrieve/pii/S0378720614001116|journal=Information &amp; Management|language=en|volume=52|issue=2|pages=183–199|doi=10.1016/j.im.2014.08.008}}&lt;/ref&gt;. After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : 
{| class=&quot;wikitable&quot;
|+
!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 &quot;{{Wikidata entity link|Q137901202}}&quot; 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}}, &quot;energy transition&quot; 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 &quot;filter&quot; 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 :&lt;ref&gt;{{Cite journal|last=Nielsen|first=Finn Årup|last2=Lyhne|first2=Ivar|last3=Garigliotti|first3=Dario|last4=Butzbach|first4=Annika|last5=Ravn Boess|first5=Emilia|last6=Hose|first6=Katja|last7=Kørnøv|first7=Lone|date=2023|title=Environmental impact assessment reports in Wikidata and a Wikibase|url=https://repositum.tuwien.at/handle/20.500.12708/193492|language=en|publisher=CEUR-WS.org|volume=3443|pages=1–8|doi=10.34726/5421}}&lt;/ref&gt;)  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)&lt;ref name=&quot;:19&quot;&gt;Cañas, Alberto J., et al. &quot;CmapTools: A knowledge modeling and sharing environment.&quot; (2004): 125-135. https://thomaseskridge.com/assets/pdf/Canas-2004.pdf&lt;/ref&gt;. Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]&lt;ref name=&quot;:19&quot; /&gt;, Altas.Ti &quot;network&quot; feature...). The &quot;box and arrow&quot; 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”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. 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&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[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.&quot;&lt;ref name=&quot;:21&quot;&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; After identifying themes, qualitative researchers are expected to theorize the &quot;arrows&quot; between themes&lt;ref name=&quot;:21&quot; /&gt;R.esearchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;. 

=== 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 &quot;concept&quot; 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&lt;ref name=&quot;:22&quot; /&gt; 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...&lt;ref name=&quot;:22&quot;&gt;{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}&lt;/ref&gt;, and scholars have advised to take insight from philosophy to work on concepts&lt;ref&gt;{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}&lt;/ref&gt;. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt;. 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 &quot;something&quot; 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 &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, 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&amp;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 &quot;does not have effect&quot; seems contrary to what is listed in &quot;has goal&quot;.) We would however argue this is not a problem because &quot;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&quot;&lt;ref&gt;{{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}}&lt;/ref&gt;. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}&lt;/ref&gt;, as long as they are supported by references&lt;ref&gt;{{Cite journal|last=Amaral|first=Gabriel|last2=Piscopo|first2=Alessandro|last3=Kaffee|first3=Lucie-aimée|last4=Rodrigues|first4=Odinaldo|last5=Simperl|first5=Elena|date=2021-12-31|title=Assessing the Quality of Sources in Wikidata Across Languages: A Hybrid Approach|url=https://dl.acm.org/doi/10.1145/3484828|journal=Journal of Data and Information Quality|language=en|volume=13|issue=4|pages=1–35|doi=10.1145/3484828|issn=1936-1955}}&lt;/ref&gt;.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, the discrepancy between the stated goals of this movement and what it actually achieves is precisely what some authors are critiquing&lt;ref name=&quot;:20&quot; /&gt;.  

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 &quot;management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served&quot;&lt;ref&gt;{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research &amp; Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}&lt;/ref&gt;, &quot;choice&quot; 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 &quot;concept&quot; and &quot;political concept&quot;.]]
===== Concision =====
Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were &quot;in the tree of another value&quot;. 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&lt;nowiki/&gt;) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using &quot;instance of&quot; as transversal property and checking &quot;instance of or subclass of&quot;).

*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 &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;&lt;ref&gt;{{Cite journal|last=Vanegas-Cantarero|first=María M.|date=2020-12-01|title=Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries|url=https://www.wikidata.org/wiki/Q137901196|journal=Energy Research &amp; Social Science|language=English|volume=70|doi=10.1016/J.ERSS.2020.101716}}&lt;/ref&gt;. Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for &quot;final energy consumption&quot;. 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&lt;ref&gt;{{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}}&lt;/ref&gt;. 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 &quot;{{Wikidata entity link|Q127270577}}&quot;). 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 &quot;natures&quot; 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)&lt;ref&gt;Fleetwood, S. (2004). An ontology for organisation and management studies. ''Critical Realist Applications in Organisation and Management Studies'', 27–53.&lt;/ref&gt;. 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 &quot;supports multiple coexisting classification&quot; and allow multiple ontological frameworks to coexist.&lt;ref name=&quot;:8&quot; /&gt; 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=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{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}}&lt;/ref&gt; 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 focused on identifying key concepts, and creating them when necessary (ex: {{Wikidata entity link|Q141223829}}, {{Wikidata entity link|Q141223998}}, {{Wikidata entity link|Q141224129}}, {{Wikidata entity link|Q141224432}}) When relevant, we included cited papers in the references along the main article we were reading.

== 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 statements using a specitic source in order to map the content of a single academic article. First test here: https://w.wiki/PFqH
*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}}).

=== Properties visualisation ===

*Visualise the &quot;tree&quot; 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 &quot;instance of&quot; values for energy democracy in Wikidata.png|center|thumb|553x553px|Visualisation of the &quot;{{Wikidata entity link|P31}}&quot; 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 &quot;topic in context&quot;)[[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 &quot;The topic in context&quot; 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] .]]

=== Mapping sources consensus ===

*Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible.

== 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&lt;ref&gt;{{Cite journal|last=Maryl|first=Maciej|last2=Blaszczyńska|first2=Marta|last3=Zalotyńska|first3=Agnieszka|last4=Taylor|first4=Laurence|last5=Avanço|first5=Karla|last6=Balula|first6=Ana|last7=Buchner|first7=Anna|last8=Caliman|first8=Lorena|last9=Clivaz|first9=Claire|date=2021-01|title=Future of Scholarly Communication|url=https://hal.science/hal-03277615}}&lt;/ref&gt;). 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&lt;ref&gt;{{Cite web|url=https://eprints.whiterose.ac.uk/id/eprint/226771/|title=CLOUDSENSE Collaborative Review of Wikipedia Cloud Feedback article|last=Finney|first=D.|last2=Huang|first2=X.|date=2025-05-16|website=eprints.whiterose.ac.uk|language=en|doi=10.48785/100/329|access-date=2026-07-28|last3=Hinnie|first3=L.|last4=Sheppard|first4=N.}}&lt;/ref&gt;.

=== 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 &quot;remixed&quot; 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&lt;ref&gt;{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}&lt;/ref&gt;. Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to &quot;navigate&quot; 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 &quot;navigate&quot; 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&lt;ref name=&quot;:9&quot; /&gt;. 

=== Quality assessment ===
&lt;ref&gt;{{Citation|vauthors=((E. Peireira, A.))|year=2022|title=Usages et transformations d’un commun numérique : les données généalogiques équines|publisher=Paris 10|url=https://theses.fr/s357002|access-date=25 August 2026}}&lt;/ref&gt;  

=== Future research ===
- Can we &quot;reason&quot; 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 &lt;ref name=&quot;:24&quot;&gt;{{cite journal|date=June 2015|title=Novices’ Struggles with Conceptual and Theoretical Framing in Writing Dissertations and Papers for Publication|journal=Publications|publisher=Multidisciplinary Digital Publishing Institute|volume=3|issue=2|pages=104–119|doi=10.3390/publications3020104|issn=2304-6775|vauthors=((Casanave, C. P.)), ((Li, Y.))}}&lt;/ref&gt;. Conceptualization require to answer the question « what are our data examples of? » &lt;ref name=&quot;:24&quot; /&gt;.  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 &lt;ref name=&quot;:24&quot; /&gt;. 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_&amp;_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=&quot;wikitable sortable&quot;
! 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}}</text>
      <sha1>h29tbo1371j59rnxhbhtmxgyl1eu0rr</sha1>
    </revision>
    <revision>
      <id>2829782</id>
      <parentid>2829781</parentid>
      <timestamp>2026-08-30T22:07:22Z</timestamp>
      <contributor>
        <username>Jeanne Noiraud</username>
        <id>1366702</id>
      </contributor>
      <comment>/* Knowledge modelling in Wikidata : second round of analysis */ mentionning concept created</comment>
      <origin>2829782</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="115147" sha1="t3vp7h6qukubca0j76v7fm61tisfd66" xml:space="preserve">== Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) ==

== Acknowledgements ==
The present text was originally written on a Wikiversity page, if you are reading it in another format, you can find this page here : [[Just sustainability transitions: a living review|https://en.wikiversity.org/wiki/Just_sustainability_transitions:_a_living_review]]. You are free to add your comments on the page discussion section. 

=== Contributors ===
{| class=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-3-030-89460-3_2|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021|publisher=Springer International Publishing|isbn=978-3-030-89460-3|editor-last=Heffron|editor-first=Raphael J.|location=Cham|pages=9–19|language=en|doi=10.1007/978-3-030-89460-3_2}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.sciencedirect.com/science/article/pii/S0301421518302301|journal=Energy Policy|volume=119|pages=1–7|doi=10.1016/j.enpol.2018.04.014|issn=0301-4215}}&lt;/ref&gt;. 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&lt;ref name=&quot;:15&quot;&gt;{{Cite journal|date=2020-03-01|title=Micro-foundations of the multi-level perspective on socio-technical transitions: Developing a multi-dimensional model of agency through crossovers between social constructivism, evolutionary economics and neo-institutional theory|url=https://www.sciencedirect.com/science/article/abs/pii/S0040162518316111|journal=Technological Forecasting and Social Change|language=en-US|volume=152|pages=119894|doi=10.1016/j.techfore.2019.119894|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2023-08-01|title=A socio-technical transition perspective on positive tipping points in climate change mitigation: Analysing seven interacting feedback loops in offshore wind and electric vehicles acceleration|url=https://www.sciencedirect.com/science/article/pii/S0040162523003244|journal=Technological Forecasting and Social Change|language=en-US|volume=193|pages=122639|doi=10.1016/j.techfore.2023.122639|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Sovacool|first=Benjamin K.|last2=Geels|first2=Frank W.|last3=Andersen|first3=Allan Dahl|last4=Grubb|first4=Michael|last5=Jordan|first5=Andrew J.|last6=Kern|first6=Florian|last7=Kivimaa|first7=Paula|last8=Lockwood|first8=Matthew|last9=Markard|first9=Jochen|date=2025-03-01|title=The acceleration of low-carbon transitions: Insights, concepts, challenges, and new directions for research|url=https://www.sciencedirect.com/science/article/pii/S2214629625000295|journal=Energy Research &amp; Social Science|volume=121|pages=103948|doi=10.1016/j.erss.2025.103948|issn=2214-6296}}&lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Kepes|first=Sven|last2=Bennett|first2=Andrew A.|last3=McDaniel|first3=Michael A.|date=2014-09|title=Evidence-Based Management and the Trustworthiness of Our Cumulative Scientific Knowledge: Implications for Teaching, Research, and Practice|url=https://journals.aom.org/doi/10.5465/amle.2013.0193|journal=Academy of Management Learning &amp; Education|volume=13|issue=3|pages=446–466|doi=10.5465/amle.2013.0193|issn=1537-260X}}&lt;/ref&gt;&lt;ref&gt;Pfeffer, J., &amp; Sutton, R. I. (2006). Evidence-Based Management. Harvard Business Review, 13. &lt;/ref&gt;. 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&lt;ref&gt;{{Cite journal|last=Bornmann|first=Lutz|last2=Mutz|first2=Rüdiger|date=2015|title=Growth rates of modern science: A bibliometric analysis based on the number of publications and cited references|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/asi.23329|journal=Journal of the Association for Information Science and Technology|language=en|volume=66|issue=11|pages=2215–2222|doi=10.1002/asi.23329|issn=2330-1643}}&lt;/ref&gt;&lt;ref name=&quot;:23&quot;&gt;{{Cite journal|last=Hanson|first=Mark A.|last2=Barreiro|first2=Pablo Gómez|last3=Crosetto|first3=Paolo|last4=Brockington|first4=Dan|date=2024-11-01|title=The strain on scientific publishing|url=https://doi.org/10.1162/qss_a_00327|journal=Quantitative Science Studies|volume=5|issue=4|pages=823–843|doi=10.1162/qss_a_00327|issn=2641-3337}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=http://blogs.nature.com/news/2014/05/global-scientific-output-doubles-every-nine-years.html|title=Global scientific output doubles every nine years : News blog|website=blogs.nature.com|language=en-US|access-date=2026-06-23}}&lt;/ref&gt;, 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&lt;ref name=&quot;:23&quot; /&gt;. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge&lt;ref&gt;{{Cite journal|last=Hartman|first=Yvonne|last2=Darab|first2=Sandy|date=2012-01-01|title=A Call for Slow Scholarship: A Case Study on the Intensification of Academic Life and Its Implications for Pedagogy|url=https://doi.org/10.1080/10714413.2012.643740|journal=Review of Education, Pedagogy, and Cultural Studies|volume=34|issue=1-2|pages=49–60|doi=10.1080/10714413.2012.643740|issn=1071-4413}}&lt;/ref&gt;. “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&lt;ref&gt;{{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)}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=L.|first=Mobley, David|last2=M.|first2=Zuckerman, Daniel|date=2015-02-03|title=A proposal for regularly updated review/survey articles: &quot;Perpetual Reviews&quot;|url=https://arxiv.org/abs/1502.01329|journal=arXiv.org|language=en|doi=10.48550/arXiv.1502.01329}}&lt;/ref&gt; and for shifting from static text format publications to dynamic knowledge mapping&lt;ref name=&quot;:11&quot;&gt;{{Cite web|url=https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/|title=The death of the literature review and the rise of the dynamic knowledge map - LSE Impact|last=Taster|date=2019-05-14|website=LSE Impact - Understanding impact and practice in academic research|access-date=2026-06-23}}&lt;/ref&gt;. This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : &lt;ref&gt;{{Cite journal|last=Elliott|first=Julian H.|last2=Synnot|first2=Anneliese|last3=Turner|first3=Tari|last4=Simmonds|first4=Mark|last5=Akl|first5=Elie A.|last6=McDonald|first6=Steve|last7=Salanti|first7=Georgia|last8=Meerpohl|first8=Joerg|last9=MacLehose|first9=Harriet|date=2017-11|title=Living systematic review: 1. Introduction—the why, what, when, and how|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435617306364|journal=Journal of Clinical Epidemiology|volume=91|pages=23–30|doi=10.1016/j.jclinepi.2017.08.010|issn=0895-4356}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Uttley|first=Lesley|last2=Quintana|first2=Daniel S.|last3=Montgomery|first3=Paul|last4=Carroll|first4=Christopher|last5=Page|first5=Matthew J.|last6=Falzon|first6=Louise|last7=Sutton|first7=Anthea|last8=Moher|first8=David|date=2023-04|title=The problems with systematic reviews: a living systematic review|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435623000112|journal=Journal of Clinical Epidemiology|volume=156|pages=30–41|doi=10.1016/j.jclinepi.2023.01.011|issn=0895-4356}}&lt;/ref&gt;,&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;). 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.

=== 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&lt;ref&gt;{{Cite journal|last=Haley|first=Usha C. V.|date=2023-09-01|title=Triviality and the Search for Scholarly Impact|url=https://doi.org/10.1177/01708406231175292|journal=Organization Studies|language=EN|volume=44|issue=9|pages=1547–1550|doi=10.1177/01708406231175292|issn=0170-8406}}&lt;/ref&gt;. Wikipedia is a key knowledge dissemination platform widely used by students&lt;ref&gt;{{Cite journal|last=Sunvy|first=Ahmed Shafkat|last2=Reza|first2=Raiyan Bin|date=2023-04-12|title=Students’ Perception of Wikipedia as an Academic Information Source|url=https://ejournal.undiksha.ac.id/index.php/IJERR/article/view/57572|journal=Indonesian Journal Of Educational Research and Review|volume=6|issue=1|pages=134–147|doi=10.23887/ijerr.v6i1.57572|issn=2621-8984}}&lt;/ref&gt; and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature&lt;ref&gt;{{Cite journal|last=Thompson|first=Neil|last2=Hanley|first2=Douglas|date=2017|title=Science Is Shaped by Wikipedia: Evidence from a Randomized Control Trial|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3039505|journal=SSRN Electronic Journal|doi=10.2139/ssrn.3039505|issn=1556-5068}}&lt;/ref&gt; and that some scholars cite directly Wikipedia&lt;ref&gt;{{Cite journal|last=Dooley|first=Patricia L.|date=2010-07-07|title=Wikipedia and the two-faced professoriate|url=https://doi.org/10.1145/1832772.1832803|journal=Proceedings of the 6th International Symposium on Wikis and Open Collaboration|series=WikiSym '10|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=1–2|doi=10.1145/1832772.1832803|isbn=978-1-4503-0056-8}}&lt;/ref&gt;. However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives&lt;ref&gt;{{Cite journal|last=Chen|first=Yan|last2=Farzan|first2=Rosta|last3=Kraut|first3=Robert|last4=YeckehZaare|first4=Iman|last5=Zhang|first5=Ark Fangzhou|date=2024-05|title=Motivating Experts to Contribute to Digital Public Goods: A Personalized Field Experiment on Wikipedia|url=https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4852|journal=Management Science|volume=70|issue=5|pages=3264–3280|doi=10.1287/mnsc.2023.4852|issn=0025-1909}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Kincaid|first=Dustin W.|last2=Beck|first2=Whitney S.|last3=Brandt|first3=Jessica E.|last4=Mars Brisbin|first4=Margaret|last5=Farrell|first5=Kaitlin J.|last6=Hondula|first6=Kelly L.|last7=Larson|first7=Erin I.|last8=Shogren|first8=Arial J.|date=2021|title=Wikipedia can help resolve information inequality in the aquatic sciences|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/lol2.10168|journal=Limnology and Oceanography Letters|language=en|volume=6|issue=1|pages=18–23|doi=10.1002/lol2.10168|issn=2378-2242}}&lt;/ref&gt;, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow&lt;ref name=&quot;:10&quot;&gt;Taraborelli, D., Mietchen, D., Alevizou, P., &amp; Gill, A. (2011, August). Expert participation on Wikipedia: Barriers and opportunities. Wikimania 2011, Haifa, Israel. &lt;nowiki&gt;http://upload.wikimedia.org/wikipedia/commons/4/4f/Expert_Participation_Survey_-_Wikimania_2011.pdf&lt;/nowiki&gt; &lt;/ref&gt;. In addition, expert participation is not immune to the gender gap&lt;ref name=&quot;:10&quot; /&gt;. Because of gender segregation in disciplines&lt;ref&gt;{{Cite journal|last=Ceci|first=Stephen J.|last2=Ginther|first2=Donna K.|last3=Kahn|first3=Shulamit|last4=Williams|first4=Wendy M.|date=2014-12-01|title=Women in Academic Science: A Changing Landscape|url=https://doi.org/10.1177/1529100614541236|journal=Psychological Science in the Public Interest|language=EN|volume=15|issue=3|pages=75–141|doi=10.1177/1529100614541236|issn=1529-1006}}&lt;/ref&gt;, this may be detrimental to the content coverage on “female” topics&lt;ref&gt;{{Cite journal|last=Lam|first=Shyong (Tony) K.|last2=Uduwage|first2=Anuradha|last3=Dong|first3=Zhenhua|last4=Sen|first4=Shilad|last5=Musicant|first5=David R.|last6=Terveen|first6=Loren|last7=Riedl|first7=John|date=2011-10-03|title=WP:clubhouse?: an exploration of Wikipedia's gender imbalance|url=https://dl.acm.org/doi/10.1145/2038558.2038560|language=en|publisher=ACM|pages=1–10|doi=10.1145/2038558.2038560|isbn=978-1-4503-0909-7}}&lt;/ref&gt;, 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 &quot;a fair and equitable process of moving towards a post-carbon society&quot;&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;. 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&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 

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&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:16&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; and fragmented results coming from various disciplines&lt;ref name=&quot;:20&quot;&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;. Then, making literature reviews &quot;living&quot; 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&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. 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&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, 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&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref name=&quot;:12&quot; /&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but not yet integrated into tested and validated methodologies&lt;ref name=&quot;:13&quot;&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;. Human validation stays notably necessary&lt;ref&gt;{{Cite journal|last=Alshami|first=Ahmad|last2=Elsayed|first2=Moustafa|last3=Ali|first3=Eslam|last4=Eltoukhy|first4=Abdelrahman E. E.|last5=Zayed|first5=Tarek|date=2023-07-09|title=Harnessing the Power of ChatGPT for Automating Systematic Review Process: Methodology, Case Study, Limitations, and Future Directions|url=https://www.mdpi.com/2079-8954/11/7/351|journal=Systems|language=en|volume=11|issue=7|pages=351|doi=10.3390/systems11070351|issn=2079-8954}}&lt;/ref&gt;,&lt;ref name=&quot;:13&quot; /&gt;. 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&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&amp;oldid=1360514388|journal=Wikipedia|language=en}}&lt;/ref&gt;, 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, and a &quot;collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation&quot;.&lt;ref&gt;{{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}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&amp;oldid=1360462340|journal=Wikipedia|language=en}}&lt;/ref&gt; &quot;A [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}&lt;/ref&gt; 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.&quot;&lt;ref&gt;{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}&lt;/ref&gt; Such graphs have a strong potential to conduct knowledge synthesis&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;&lt;ref name=&quot;:18&quot; /&gt;. They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;.  

Wikidata is multilingual... 

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]])&lt;ref&gt;{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&amp;oldid=1358552930|journal=Wikipedia|language=en}}&lt;/ref&gt;. Like all Wikimedia projects, Wikiversity is editable by everyone, have a discussion tab and a history log tab. 

=== Use of Wikidata and Wikiversity for research ===

== 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&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt; (e.g. the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;). 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)&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 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=&quot;wikitable&quot;
|+
!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 &quot;export to Wikidata QuickStatements&quot; 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|&quot;Merge&quot; 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. &lt;ref name=&quot;:5&quot; /&gt;). 

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=&quot;wikitable&quot;
|+
!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}}&lt;ref&gt;{{Cite journal|last=Paré|first=Guy|last2=Trudel|first2=Marie-Claude|last3=Jaana|first3=Mirou|last4=Kitsiou|first4=Spyros|date=2015-03|title=Synthesizing information systems knowledge: A typology of literature reviews|url=https://linkinghub.elsevier.com/retrieve/pii/S0378720614001116|journal=Information &amp; Management|language=en|volume=52|issue=2|pages=183–199|doi=10.1016/j.im.2014.08.008}}&lt;/ref&gt;. After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : 
{| class=&quot;wikitable&quot;
|+
!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 &quot;{{Wikidata entity link|Q137901202}}&quot; 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}}, &quot;energy transition&quot; 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 &quot;filter&quot; 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 :&lt;ref&gt;{{Cite journal|last=Nielsen|first=Finn Årup|last2=Lyhne|first2=Ivar|last3=Garigliotti|first3=Dario|last4=Butzbach|first4=Annika|last5=Ravn Boess|first5=Emilia|last6=Hose|first6=Katja|last7=Kørnøv|first7=Lone|date=2023|title=Environmental impact assessment reports in Wikidata and a Wikibase|url=https://repositum.tuwien.at/handle/20.500.12708/193492|language=en|publisher=CEUR-WS.org|volume=3443|pages=1–8|doi=10.34726/5421}}&lt;/ref&gt;)  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)&lt;ref name=&quot;:19&quot;&gt;Cañas, Alberto J., et al. &quot;CmapTools: A knowledge modeling and sharing environment.&quot; (2004): 125-135. https://thomaseskridge.com/assets/pdf/Canas-2004.pdf&lt;/ref&gt;. Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]&lt;ref name=&quot;:19&quot; /&gt;, Altas.Ti &quot;network&quot; feature...). The &quot;box and arrow&quot; 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”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. 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&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[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.&quot;&lt;ref name=&quot;:21&quot;&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; After identifying themes, qualitative researchers are expected to theorize the &quot;arrows&quot; between themes&lt;ref name=&quot;:21&quot; /&gt;R.esearchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;. 

=== 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 &quot;concept&quot; 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&lt;ref name=&quot;:22&quot; /&gt; 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...&lt;ref name=&quot;:22&quot;&gt;{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}&lt;/ref&gt;, and scholars have advised to take insight from philosophy to work on concepts&lt;ref&gt;{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}&lt;/ref&gt;. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt;. 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 &quot;something&quot; 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 &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, 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&amp;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 &quot;does not have effect&quot; seems contrary to what is listed in &quot;has goal&quot;.) We would however argue this is not a problem because &quot;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&quot;&lt;ref&gt;{{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}}&lt;/ref&gt;. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}&lt;/ref&gt;, as long as they are supported by references&lt;ref&gt;{{Cite journal|last=Amaral|first=Gabriel|last2=Piscopo|first2=Alessandro|last3=Kaffee|first3=Lucie-aimée|last4=Rodrigues|first4=Odinaldo|last5=Simperl|first5=Elena|date=2021-12-31|title=Assessing the Quality of Sources in Wikidata Across Languages: A Hybrid Approach|url=https://dl.acm.org/doi/10.1145/3484828|journal=Journal of Data and Information Quality|language=en|volume=13|issue=4|pages=1–35|doi=10.1145/3484828|issn=1936-1955}}&lt;/ref&gt;.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, the discrepancy between the stated goals of this movement and what it actually achieves is precisely what some authors are critiquing&lt;ref name=&quot;:20&quot; /&gt;.  

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 &quot;management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served&quot;&lt;ref&gt;{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research &amp; Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}&lt;/ref&gt;, &quot;choice&quot; 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 &quot;concept&quot; and &quot;political concept&quot;.]]
===== Concision =====
Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were &quot;in the tree of another value&quot;. 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&lt;nowiki/&gt;) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using &quot;instance of&quot; as transversal property and checking &quot;instance of or subclass of&quot;).

*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 &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;&lt;ref&gt;{{Cite journal|last=Vanegas-Cantarero|first=María M.|date=2020-12-01|title=Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries|url=https://www.wikidata.org/wiki/Q137901196|journal=Energy Research &amp; Social Science|language=English|volume=70|doi=10.1016/J.ERSS.2020.101716}}&lt;/ref&gt;. Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for &quot;final energy consumption&quot;. 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&lt;ref&gt;{{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}}&lt;/ref&gt;. 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 &quot;{{Wikidata entity link|Q127270577}}&quot;). 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 &quot;natures&quot; 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)&lt;ref&gt;Fleetwood, S. (2004). An ontology for organisation and management studies. ''Critical Realist Applications in Organisation and Management Studies'', 27–53.&lt;/ref&gt;. 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 &quot;supports multiple coexisting classification&quot; and allow multiple ontological frameworks to coexist.&lt;ref name=&quot;:8&quot; /&gt; 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=&quot;wikitable&quot;
|+
!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&lt;ref&gt;{{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}}&lt;/ref&gt; 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 focused on identifying key concepts, and creating them when necessary (ex: {{Wikidata entity link|Q141223829}}, {{Wikidata entity link|Q141223998}}, {{Wikidata entity link|Q141224129}}, {{Wikidata entity link|Q141224432}}, {{Wikidata entity link|Q141224695}}) When relevant, we included cited papers in the references along the main article we were reading.

== 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 statements using a specitic source in order to map the content of a single academic article. First test here: https://w.wiki/PFqH
*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}}).

=== Properties visualisation ===

*Visualise the &quot;tree&quot; 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 &quot;instance of&quot; values for energy democracy in Wikidata.png|center|thumb|553x553px|Visualisation of the &quot;{{Wikidata entity link|P31}}&quot; 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 &quot;topic in context&quot;)[[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 &quot;The topic in context&quot; 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] .]]

=== Mapping sources consensus ===

*Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible.

== 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&lt;ref&gt;{{Cite journal|last=Maryl|first=Maciej|last2=Blaszczyńska|first2=Marta|last3=Zalotyńska|first3=Agnieszka|last4=Taylor|first4=Laurence|last5=Avanço|first5=Karla|last6=Balula|first6=Ana|last7=Buchner|first7=Anna|last8=Caliman|first8=Lorena|last9=Clivaz|first9=Claire|date=2021-01|title=Future of Scholarly Communication|url=https://hal.science/hal-03277615}}&lt;/ref&gt;). 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&lt;ref&gt;{{Cite web|url=https://eprints.whiterose.ac.uk/id/eprint/226771/|title=CLOUDSENSE Collaborative Review of Wikipedia Cloud Feedback article|last=Finney|first=D.|last2=Huang|first2=X.|date=2025-05-16|website=eprints.whiterose.ac.uk|language=en|doi=10.48785/100/329|access-date=2026-07-28|last3=Hinnie|first3=L.|last4=Sheppard|first4=N.}}&lt;/ref&gt;.

=== 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 &quot;remixed&quot; 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&lt;ref&gt;{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}&lt;/ref&gt;. Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to &quot;navigate&quot; 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 &quot;navigate&quot; 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&lt;ref name=&quot;:9&quot; /&gt;. 

=== Quality assessment ===
&lt;ref&gt;{{Citation|vauthors=((E. Peireira, A.))|year=2022|title=Usages et transformations d’un commun numérique : les données généalogiques équines|publisher=Paris 10|url=https://theses.fr/s357002|access-date=25 August 2026}}&lt;/ref&gt;  

=== Future research ===
- Can we &quot;reason&quot; 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 &lt;ref name=&quot;:24&quot;&gt;{{cite journal|date=June 2015|title=Novices’ Struggles with Conceptual and Theoretical Framing in Writing Dissertations and Papers for Publication|journal=Publications|publisher=Multidisciplinary Digital Publishing Institute|volume=3|issue=2|pages=104–119|doi=10.3390/publications3020104|issn=2304-6775|vauthors=((Casanave, C. P.)), ((Li, Y.))}}&lt;/ref&gt;. Conceptualization require to answer the question « what are our data examples of? » &lt;ref name=&quot;:24&quot; /&gt;.  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 &lt;ref name=&quot;:24&quot; /&gt;. 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_&amp;_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=&quot;wikitable sortable&quot;
! 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}}</text>
      <sha1>t3vp7h6qukubca0j76v7fm61tisfd66</sha1>
    </revision>
  </page>
  <page>
    <title>DesignWriteStudio/Course/Assignments/Group4/4.1 Rename Your Pages</title>
    <ns>0</ns>
    <id>327951</id>
    <revision>
      <id>2829867</id>
      <parentid>2793722</parentid>
      <timestamp>2026-08-31T10:07:09Z</timestamp>
      <contributor>
        <username>WAIVISA</username>
        <id>3110349</id>
      </contributor>
      <comment>/* Objective */</comment>
      <origin>2829867</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9492" sha1="jrao87y10j4pr3bt6b8h106cgfpmdg3" xml:space="preserve">{{:DesignWriteStudio/SiteElements/Navbox}}

== Assignment 4.1: Rename Your Pages ==

```html
&lt;!DOCTYPE html&gt;
&lt;html lang=&quot;en&quot;&gt;
&lt;head&gt;
  &lt;meta charset=&quot;UTF-8&quot;&gt;
  &lt;meta name=&quot;viewport&quot; content=&quot;width=device-width, initial-scale=1.0&quot;&gt;
  &lt;title&gt;WAIVI MAP&lt;/title&gt;

  &lt;style&gt;
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: Arial, sans-serif;
    }

    body {
      background: #f4f7fb;
      color: #222;
    }

    header {
      background: #111827;
      color: white;
      padding: 18px 25px;
      display: flex;
      align-items: center;
      justify-content: space-between;
      position: relative;
      z-index: 1000;
    }

    .logo {
      font-size: 28px;
      font-weight: bold;
      letter-spacing: 2px;
    }

    .logo span {
      color: #00d4ff;
    }

    nav a {
      color: white;
      text-decoration: none;
      margin-left: 20px;
      font-size: 15px;
    }

    nav a:hover {
      color: #00d4ff;
    }

    .hero {
      background: linear-gradient(135deg, #111827, #1e3a8a);
      color: white;
      padding: 45px 20px;
      text-align: center;
    }

    .hero h1 {
      font-size: 45px;
      margin-bottom: 10px;
    }

    .hero p {
      font-size: 18px;
      opacity: 0.9;
      margin-bottom: 25px;
    }

    .search-box {
      max-width: 650px;
      margin: auto;
      display: flex;
      background: white;
      border-radius: 50px;
      padding: 6px;
    }

    .search-box input {
      flex: 1;
      border: none;
      outline: none;
      padding: 15px 20px;
      font-size: 16px;
      border-radius: 50px;
    }

    .search-box button {
      border: none;
      background: #00a8ff;
      color: white;
      padding: 0 25px;
      border-radius: 40px;
      cursor: pointer;
      font-weight: bold;
    }

    .search-box button:hover {
      background: #0086cc;
    }

    #map {
      height: 550px;
      width: 100%;
      border-bottom: 4px solid #00a8ff;
    }

    .locations {
      padding: 40px 20px;
      max-width: 1100px;
      margin: auto;
    }

    .locations h2 {
      text-align: center;
      margin-bottom: 25px;
      font-size: 30px;
    }

    .cards {
      display: grid;
      grid-template-columns: repeat(auto-fit, minmax(220px, 1fr));
      gap: 20px;
    }

    .card {
      background: white;
      padding: 25px;
      border-radius: 15px;
      box-shadow: 0 5px 20px rgba(0,0,0,0.08);
      transition: 0.3s;
    }

    .card:hover {
      transform: translateY(-5px);
    }

    .card h3 {
      margin-bottom: 10px;
      color: #1e3a8a;
    }

    .card p {
      color: #666;
      margin-bottom: 15px;
    }

    .card button {
      border: none;
      background: #111827;
      color: white;
      padding: 10px 16px;
      border-radius: 8px;
      cursor: pointer;
    }

    footer {
      background: #111827;
      color: white;
      text-align: center;
      padding: 25px;
      margin-top: 30px;
    }

    @media(max-width: 600px) {
      .hero h1 {
        font-size: 32px;
      }

      nav {
        display: none;
      }

      #map {
        height: 450px;
      }
    }
  &lt;/style&gt;
&lt;/head&gt;

&lt;body&gt;

&lt;header&gt;
  &lt;div class=&quot;logo&quot;&gt;WAIVI&lt;span&gt;MAP&lt;/span&gt;&lt;/div&gt;

  &lt;nav&gt;
    &lt;a href=&quot;#home&quot;&gt;Home&lt;/a&gt;
    &lt;a href=&quot;#map&quot;&gt;Map&lt;/a&gt;
    &lt;a href=&quot;#locations&quot;&gt;Locations&lt;/a&gt;
  &lt;/nav&gt;
&lt;/header&gt;

&lt;section class=&quot;hero&quot; id=&quot;home&quot;&gt;
  &lt;h1&gt;WAIVI MAP&lt;/h1&gt;
  &lt;p&gt;Explore Rwanda. Discover places. Find your way.&lt;/p&gt;

  &lt;div class=&quot;search-box&quot;&gt;
    &lt;input
      type=&quot;text&quot;
      id=&quot;searchInput&quot;
      placeholder=&quot;Search Kigali, Musanze, Huye...&quot;
    &gt;
    &lt;button onclick=&quot;searchLocation()&quot;&gt;Search&lt;/button&gt;
  &lt;/div&gt;
&lt;/section&gt;

&lt;div id=&quot;map&quot;&gt;&lt;/div&gt;

&lt;section class=&quot;locations&quot; id=&quot;locations&quot;&gt;

  &lt;h2&gt;Popular Rwanda Locations&lt;/h2&gt;

  &lt;div class=&quot;cards&quot;&gt;

    &lt;div class=&quot;card&quot;&gt;
      &lt;h3&gt;📍 Kigali&lt;/h3&gt;
      &lt;p&gt;The capital city of Rwanda.&lt;/p&gt;
      &lt;button onclick=&quot;goToLocation(-1.9441, 30.0619)&quot;&gt;
        View on Map
      &lt;/button&gt;
    &lt;/div&gt;

    &lt;div class=&quot;card&quot;&gt;
      &lt;h3&gt;🌋 Musanze&lt;/h3&gt;
      &lt;p&gt;Gateway to Rwanda's famous volcanoes.&lt;/p&gt;
      &lt;button onclick=&quot;goToLocation(-1.4998, 29.6347)&quot;&gt;
        View on Map
      &lt;/button&gt;
    &lt;/div&gt;

    &lt;div class=&quot;card&quot;&gt;
      &lt;h3&gt;🌿 Huye&lt;/h3&gt;
      &lt;p&gt;A major cultural and educational city.&lt;/p&gt;
      &lt;button onclick=&quot;goToLocation(-2.5967, 29.7394)&quot;&gt;
        View on Map
      &lt;/button&gt;
    &lt;/div&gt;

    &lt;div class=&quot;card&quot;&gt;
      &lt;h3&gt;🌊 Rubavu&lt;/h3&gt;
      &lt;p&gt;Beautiful city on Lake Kivu.&lt;/p&gt;
      &lt;button onclick=&quot;goToLocation(-1.6773, 29.2620)&quot;&gt;
        View on Map
      &lt;/button&gt;
    &lt;/div&gt;

  &lt;/div&gt;
&lt;/section&gt;

&lt;footer&gt;
  &lt;p&gt;© 2026 WAIVI MAP — Explore Rwanda 🇷🇼&lt;/p&gt;
  &lt;p&gt;Created by WAIVI THE BOSS&lt;/p&gt;
&lt;/footer&gt;

&lt;!-- Leaflet Map --&gt;
&lt;link
  rel=&quot;stylesheet&quot;
  href=&quot;https://unpkg.com/leaflet@1.9.4/dist/leaflet.css&quot;
/&gt;

&lt;script src=&quot;https://unpkg.com/leaflet@1.9.4/dist/leaflet.js&quot;&gt;&lt;/script&gt;

&lt;script&gt;

  // Create map centered on Rwanda
  const map = L.map('map').setView([-1.9403, 29.8739], 8);

  // Add OpenStreetMap tiles
  L.tileLayer(
    'https://{s}.tile.openstreetmap.org/{z}/{x}/{y}.png',
    {
      maxZoom: 19,
      attribution: '&amp;copy; OpenStreetMap contributors'
    }
  ).addTo(map);

  // Locations
  const locations = [
    {
      name: &quot;Kigali&quot;,
      lat: -1.9441,
      lng: 30.0619
    },
    {
      name: &quot;Musanze&quot;,
      lat: -1.4998,
      lng: 29.6347
    },
    {
      name: &quot;Huye&quot;,
      lat: -2.5967,
      lng: 29.7394
    },
    {
      name: &quot;Rubavu&quot;,
      lat: -1.6773,
      lng: 29.2620
    }
  ];

  // Add markers
  locations.forEach(function(location) {

    L.marker([location.lat, location.lng])
      .addTo(map)
      .bindPopup(
        &quot;&lt;b&gt;WAIVI MAP&lt;/b&gt;&lt;br&gt;&quot; +
        location.name +
        &quot;&lt;br&gt;&lt;br&gt;🇷🇼 Rwanda&quot;
      );

  });

  // Move map to selected location
  function goToLocation(lat, lng) {

    map.setView([lat, lng], 13);

  }

  // Search location
  function searchLocation() {

    const search =
      document.getElementById(&quot;searchInput&quot;)
      .value
      .toLowerCase()
      .trim();

    const result = locations.find(function(location) {

      return location.name.toLowerCase() === search;

    });

    if (result) {

      map.setView(
        [result.lat, result.lng],
        13
      );

      L.popup()
        .setLatLng([result.lat, result.lng])
        .setContent(
          &quot;&lt;b&gt;WAIVI MAP&lt;/b&gt;&lt;br&gt;&quot; +
          result.name +
          &quot;&lt;br&gt;🇷🇼 Rwanda&quot;
        )
        .openOn(map);

    } else {

      alert(
        &quot;Location not found. Try Kigali, Musanze, Huye or Rubavu.&quot;
      );

    }
  }

&lt;/script&gt;

&lt;/body&gt;
&lt;/html&gt;
```

=== Questions? ===
Post to the '''Discuss''' tab of this page. Questions are answered there so all students — including async — can see the response. Do not email or use Discord for assignment questions.

=== The Rule: Move, Don't Recreate ===
MediaWiki has a '''Move''' function for exactly this purpose. Use it. Do not copy-paste content into a new page and delete the old one.

Why it matters:
* '''Move''' preserves full edit history — every revision stays attached to the page
* '''Move''' creates an automatic redirect from the old title, so any existing links keep working
* '''Move''' preserves the Talk page
* Recreating breaks all of the above

=== Standard Naming Convention ===
Your pages should follow this pattern:

&lt;code&gt;DesignWriteStudio/Course/StudentPages/YourName/X.X Title&lt;/code&gt;

Where &lt;code&gt;X.X&lt;/code&gt; is the assignment number and &lt;code&gt;Title&lt;/code&gt; is a short descriptive title. Examples:

{| class=&quot;wikitable&quot;
|-
! Old name !! New name
|-
| &lt;code&gt;StudentPages/Ckrauza/hyper&lt;/code&gt; || &lt;code&gt;StudentPages/Ckrauza/1.1 Defining Hypertext&lt;/code&gt;
|-
| &lt;code&gt;StudentPages/MarysonLazer/HypertextInfo&lt;/code&gt; || &lt;code&gt;StudentPages/MarysonLazer/1.1 Defining Hypertext&lt;/code&gt;
|-
| &lt;code&gt;StudentPages/PaulLee/hyper&lt;/code&gt; || &lt;code&gt;StudentPages/PaulLee/1.1 Defining Hypertext&lt;/code&gt;
|}

Use your own student name and the assignment number and title that matches the content of each page.

=== How to Move a Page ===
# Navigate to the page you want to rename
# Click the '''More''' menu (top of page) → '''Move'''
# In the '''To new title''' field, type the full new page name
# Leave '''Leave a redirect behind''' checked — this is important
# Edit summary: &lt;code&gt;4.1 Rename: moved to standard naming convention&lt;/code&gt;
# Click '''Move page'''

Repeat for each page you created in Groups 1–3.

=== Required Deliverables ===
# All your Group 1–3 work pages moved to the standard convention
# Your student dashboard updated to reflect the new page names (edit the links in your dashboard)
# Edit summary on each move must read: &lt;code&gt;4.1 Rename: moved to standard naming convention&lt;/code&gt;

No AI transcript required for this assignment. This is an infrastructure task, not a research task.

=== Check Your Work ===
After moving, visit your student dashboard. Two things should happen automatically:

* The '''My pages''' section (powered by &lt;code&gt;Special:PrefixIndex&lt;/code&gt;) will list your renamed pages
* The '''My assignment checklist''' links — which were built using the standard naming convention — will turn from red (missing) to blue (live)

Your dashboard was designed anticipating these names. Completing 4.1 activates it. From this point on, click a checklist link to start each new assignment page directly from your dashboard.

{{:DesignWriteStudio/SiteElements/Footer}}
[[Category:DesignWriteStudio]]</text>
      <sha1>jrao87y10j4pr3bt6b8h106cgfpmdg3</sha1>
    </revision>
  </page>
  <page>
    <title>Athena problem</title>
    <ns>0</ns>
    <id>329548</id>
    <revision>
      <id>2829785</id>
      <parentid>2829163</parentid>
      <timestamp>2026-08-30T22:20:45Z</timestamp>
      <contributor>
        <username>Athene241</username>
        <id>3100061</id>
      </contributor>
      <comment>/* Solve the problem */</comment>
      <origin>2829785</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="157161" sha1="b5qxyko9rx5rozpxc4sxgv3k9lgd7h9" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1 (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s), this set is called '''Athena set''', and the prime numbers in this set are called '''Athena primes'''.

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the Athena set in every base ''b'' must be a [[:w:Finite set|finite set]], and every base ''b'' ≥ 2 has only finitely many Athena primes, e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), the Athena set has exactly 77 [[:w:Element of a set|element]]s (they are exactly the Athena primes in decimal (base ''b'' = 10)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}.

Determining the set of the minimal elements of a arbitrary set of strings under the subsequence ordering is in general [[:w:List of unsolved problems in mathematics|unsolvable]], and can be difficult even when this set is relatively simple (such as the base ''b'' representations of the prime numbers &gt; ''b'').

Although the set ''M''(''S'') of minimal strings is necessarily [[:w:Finite set|finite]], determining it explicitly for a given ''S'' can be a difficult computational problem. We use some [[:w:Number theory|numbertheoretic]] [[:w:Heuristic argument|heuristic]]s to [[:w:Computing|compute]] ''M''(''L''&lt;sub&gt;''b''&lt;/sub&gt;), where ''L''&lt;sub&gt;''b''&lt;/sub&gt; is the [[:w:Formal language|language]] of [[:w:Radix|base]]-''b'' representations of the [[:w:Prime number|prime number]]s which are [[:w:Greater than|&gt;]] ''b'', for 2 ≤ ''b'' ≤ 36.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of Athena primes in base ''b''&quot; and &quot;length of the largest Athena prime in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the Athena set in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem (i.e. to compute the Athena set), we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist) (in fact, it is known that exactly what bases 2 ≤ ''b'' ≤ 1024 have the families listed in the table below as unsolved families, all of these families in all bases 2 ≤ ''b'' ≤ 1024 have been searched to length ≥ 10000 (for the family (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1), bases 2 ≤ ''b'' ≤ 1048576, searched to length ≥ 5000)): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)):

'''Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?'''

An [[:w:Algorithm|algorithm]] to solve this problem, for example, would allow us to decide if there are any additional [[:w:Fermat prime|Fermat prime]]s other than the known ones (corresponding to ''n'' = 0, 1, 2, 3, 4). To see this, take ''b'' = 2, ''x'' = 1, ''y'' = 0, and ''z'' = 0&lt;sub&gt;16&lt;/sub&gt;1. Since if 2&lt;sup&gt;''n''&lt;/sup&gt;+1 is prime then ''n'' must be a [[:w:Power of 2|power of two]], a prime of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' in base ''b'' must be a new Fermat prime. Besides, it would allow us to decide if there are infinitely many [[:w:Mersenne prime|Mersenne prime]]s (of the form 2&lt;sup&gt;''p''&lt;/sup&gt;−1 with prime ''p''). To see this, take ''b'' = 2, ''x'' = ''𝜆'' (the [[:w:Empty string|empty string]]), ''y'' = 1, and ''z'' = 1&lt;sub&gt;''n''+1&lt;/sub&gt;, where ''n'' is the exponent of the Mersenne prime which we want to know whether it is the largest Mersenne prime or not. Since if 2&lt;sup&gt;''n''&lt;/sup&gt;−1 is prime then ''n'' must be a [[:w:Prime number|prime]], a prime of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' in base ''b'' must be a new Mersenne prime. Also, it would allow us to decide whether 78557 is the smallest [[:w:Sierpinski number|Sierpinski number]] (i.e. odd numbers ''k'' such that ''k''×2&lt;sup&gt;''n''&lt;/sup&gt;+1 is composite for all ''n'' ≥ 1) and whether 509203 is the smallest [[:w:Riesel number|Riesel number]] (i.e. odd numbers ''k'' such that ''k''×2&lt;sup&gt;*n*&lt;/sup&gt;−1 is composite for all ''n'' ≥ 1), etc.

'''Athena conjecture (which is very important for the Athena problem): If family ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' (with fixed strings ''x'', ''z'' (may be empty), fixed digit ''y'', and variable ''n'') in base ''b'' (with fixed ''b'' ≥ 2) (''x'' does not start with the digit 0, ''z'' ends with a digit which coprime to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty) cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then family ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' in base ''b'' contains infinitely many primes (this is equivalent to: If form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) (with fixed integers ''a'' ≥ 1, ''b'' ≥ 2, ''c'' ≠ 0 (with ''gcd''(''a'',''c'') = 1 and ''gcd''(''b'',''c'') = 1), and variable ''n'') cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) contains infinitely many primes).'''

(in fact, the Athena conjecture is equivalent to the conjecture (to prove this, by change the base (''b'') to a power of ''b'' which is larger than the largest prime in a given family (in base ''b'') which only contains finitely many primes): If family ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' (with fixed strings ''x'', ''z'' (may be empty), fixed digit ''y'', and variable ''n'') in base ''b'' (with fixed ''b'' ≥ 2) (''x'' does not start with the digit 0, ''z'' ends with a digit which coprime to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty) cannot be proven to only contain composites (by covering congruence, algebraic factorization, or combine of them), then family ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' in base ''b'' contains at least one prime (this is equivalent to: If form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) (with fixed integers ''a'' ≥ 1, ''b'' ≥ 2, ''c'' ≠ 0 (with ''gcd''(''a'',''c'') = 1 and ''gcd''(''b'',''c'') = 1), and variable ''n'') cannot be proven to only contain composites (by covering congruence, algebraic factorization, or combine of them), then form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) contains at least one prime), like the [[:w:Bunyakovsky conjecture|Bunyakovsky conjecture]] and the [[:w:Dickson's conjecture|Dickson's conjecture]] and the [[:w:Schinzel's hypothesis H|Schinzel's hypothesis ''H'']], if such ''n'' always exists, then there must be always infinitely many such ''n'', to prove this, add another polynomial for the cases of the Dickson's conjecture and the Schinzel's hypothesis ''H'', also, change the polynomial (e.g. change ''n'' to ''r''×''n'' or ''n''&lt;sup&gt;''r''&lt;/sup&gt; for all integers ''r'' &gt; 1) for the cases of the Bunyakovsky conjecture and the Schinzel's hypothesis ''H'')

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 11 family 2{5}: Always divisible by 2 or 3
* The base 14 family B{0}1: Always divisible by 3 or 5
* The base 13 family 95{0}3: Always divisible by 5, 7, or 17
* The base 16 family {4}D: Always divisible by 3, 7, or 13
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family {1}: Can be written as (9&lt;sup&gt;''n''&lt;/sup&gt;−1)/8 and can be factored as (3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 8
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family 1{5}: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−1)/3 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 3
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 27 family 7{Q}: Can be written as 8×27&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (4×9&lt;sup&gt;''n''&lt;/sup&gt;+2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 27 family 9{G}: Can be written as (125×27&lt;sup&gt;''n''&lt;/sup&gt;−8)/13 and can be factored as (5×3&lt;sup&gt;''n''&lt;/sup&gt;−2) × (25×9&lt;sup&gt;''n''&lt;/sup&gt;+10×3&lt;sup&gt;''n''&lt;/sup&gt;+4)
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 14 family {D}5: Can be written as 14&lt;sup&gt;''n''&lt;/sup&gt;−9, it is divisible by 5 if ''n'' is odd and can be factored as (14&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (14&lt;sup&gt;''n''/2&lt;/sup&gt;+3) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 17 family 7{9}: Can be written as (121×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even
* The base 24 family 3{N}: Can be written as 4×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 24 family 5{N}: Can be written as 6×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is even and can be factored as (12×24&lt;sup&gt;(''n''−1)/2&lt;/sup&gt;−1) × (12×24&lt;sup&gt;(''n''−1)/2&lt;/sup&gt;+1) if ''n'' is odd

If the Athena conjecture is true, then the [[:w:Sierpiński number|Sierpiński conjecture]] and [[:w:Riesel number|Riesel conjecture]] are also true, and the [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm Sierpiński conjectures] and the [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm Riesel conjectures] in all bases ''b'' are also true, and the [http://www.noprimeleftbehind.net/crus/SNOB-Sierp-conjectures.htm real Sierpiński conjectures] and the [http://www.noprimeleftbehind.net/crus/Real-Riesel-conjectures.htm real Riesel conjectures] are also true, also, if the Athena conjecture is true, then there are infinitely many primes of these forms for fixed bases ''b'' ≥ 2 and variable exponents ''n'':

* (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) for all bases ''b'' which are not [[:w:Perfect power|perfect power]]s (for this form, ''n'' must be prime) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all even bases ''b'' which are not of the form ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r'' &gt; 1 (for this form, ''n'' must be power of 2) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101)
* (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 for all odd bases ''b'' which are not of the form ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r'' &gt; 1 (for this form, ''n'' must be power of 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt)
* (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''+1) for all bases ''b'' which are neither of the form ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r'' &gt; 1 nor of the form 4×''m''&lt;sup&gt;4&lt;/sup&gt; (for this form, ''n'' must be prime) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for prime bases ''b'': https://oeis.org/A065507)
* ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) for all bases ''b'' &gt; 2 (reference of this form: https://oeis.org/A243404)
* 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' &lt; 201446503145165177, not == 1 mod 3 (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624)
* 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591)
* 3×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all even bases ''b''
* 3×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all even bases ''b''
* 4×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' not == 1 mod 5, not == 14 mod 15, not [[:w:Fourth power|fourth power]]s
* 4×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' not == 1 mod 3, not == 4 mod 5, not [[:w:Square number|square]]s
* 5×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all even bases ''b'' &lt; 140324348, not == 1 mod 3
* 5×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all even bases ''b''
* 6×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' not == 1 mod 7, not == 34 mod 35
* 6×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' not == 1 mod 5, not == 34 mod 35, not of the form 6×''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2, 3 mod 5
* 7×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all even bases ''b''
* 7×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all even bases ''b'' &lt; 9162668342, not == 1 mod 3
* 8×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' not == 1 mod 3, not == 20 mod 21, not == 47, 83 mod 195, not == 467, 4343, 9887, 25448, 35978, 41522, 42647, 57083 mod 73815, not == 722, 83813, 206672, 239432, 322523, 1283843, 1519577, 1522553 mod 1551615, ..., not [[:w:Cube (algebra)|cube]]s
* 8×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' not == 1 mod 7, not == 20 mod 21, not == 83, 307 mod 455, not == 1266, 13593, 27292, 46353 mod 63973, ..., not [[:w:Cube (algebra)|cube]]s
* 9×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all even bases ''b'' &lt; 177744, not == 1 mod 5
* 9×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all even bases ''b'' not == 4 mod 5, not [[:w:Square number|square]]s
* 10×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' not == 1 mod 11, not == 32 mod 33 (references of this form: https://oeis.org/A088782)
* 10×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' not == 1 mod 3, not == 32 mod 33
* 11×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all even bases ''b'' not == 1 mod 3, not == 14 mod 15
* 11×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all even bases ''b'' not == 1 mod 5, not == 14 mod 15, not of the form 11×''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2, 3 mod 5
* 12×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' not == 1 mod 13, not == 142 mod 143, not == 562, 828, 900, 1166 mod 1729, not == 597, 1143 mod 1885, not == 296, 901, 1759, 3090, 4553, 5521, 5807, 6016, 6984, 7094, 7270, 7380, 7479, 8447, 8557, 8733, 8843, 9910, 10020, 10196, 10306, 11483, 11769, 12737, 14200, 15531, 16994, 18457 mod 19019, not == 563, 1433, 13212, 15097, 19848, 20718, 32497, 34382, 39133, 51782, 53667, 58418, 58452, 60337, 60883, 71067, 72952, 77737, 79622, 80168, 94267, 97022, 98583, 98907, 113552, 116307, 117868, 118192, 131967, 132513, 132837, 134398, 151252, 151798, 152122, 153683, 170537, 171083, 172968, 177753, 179638, 189822, 190368, 192253, 192287, 197038, 198923, 211572, 213568, 216323, 218208, 229987, 232853, 235608, 237493, 249272 mod 250705, ...
* 12×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' not == 1 mod 11, not == 142 mod 143, not == 307, 1143 mod 1595, not == 901, 6016, 7479, 18457 mod 19019, ...
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 for all odd bases ''b'' &lt; 201446503145165177, not == 1 mod 3 (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 for all odd bases ''b'' (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;+3 for all even bases ''b'' not divisible by 3
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;−3 for all even bases ''b'' not divisible by 3
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;+4 for all odd bases ''b'' not == 1 mod 5, not == 14 mod 15, not [[:w:Fourth power|fourth power]]s
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;−4 for all odd bases ''b'' not == 1 mod 3, not == 4 mod 5, not [[:w:Square number|square]]s
* (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139)
* (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396)
* (''b''+1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' not == 1 mod 3 (references of this form: http://www.noprimeleftbehind.net/Williams-primes-PP.htm, http://www.bitman.name/math/table/474, https://pzktupel.de/Primetables/TableWilliams4.php)
* (''b''+1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' (references of this form: http://www.noprimeleftbehind.net/Williams-primes-PM.htm, http://www.bitman.name/math/table/471, https://pzktupel.de/Primetables/TableWilliams3.php)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) for all bases ''b'' (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) for all bases ''b'' (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''+1) for all bases ''b'' not == 1 mod 3 (references of this form: http://www.bitman.name/math/table/801, https://pzktupel.de/Primetables/TableWilliams8.php, https://oeis.org/A346149, https://oeis.org/A346154)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''+1) for all bases ''b'' (references of this form: http://www.bitman.name/math/table/798, https://pzktupel.de/Primetables/TableWilliams7.php, https://oeis.org/A178250)

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored).

The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1 (the only case which [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] is [[:w:Triviality (mathematics)|trivially]] fully factored), when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run (since they are [https://t5k.org/glossary/xpage/OrdinaryPrime.html ordinary primes]). In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]].

To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1 we only used it to sieve the sequence ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'', but 2 may not divide (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)).

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates.

For the primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for the &quot;easy&quot; bases (bases ''b'' with ≤ 150 primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; (base ''b'' = 26 has 83 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 3 unsolved families, base ''b'' = 36 has 75 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 4 unsolved families, base ''b'' = 17 has 99 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 18 unsolved families, base ''b'' = 21 has 80 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 12 unsolved families, base ''b'' = 19 has 201 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation.

Currently, the final goal of the Athena problem project is finding the Athena set (i.e. finding all Athena primes) and proving that this set is exactly the Athena set (i.e. proving that these are all Athena primes (including the primality proving for the probable primes)) in all bases 2 ≤ ''b'' ≤ 36, i.e. solving all families in all bases 2 ≤ ''b'' ≤ 36. Solving all (unsolved) families in all bases 2 ≤ ''b'' ≤ 36 (and proving the primality of all probable primes in the sets of all bases 2 ≤ ''b'' ≤ 36) is not possible but we aim to solve many of them (and proving the primality of many of them), at least find a ''probable'' prime for many of them (since the smallest prime in a family may be too large (&gt; 10&lt;sup&gt;25000&lt;/sup&gt;) to be proved primality, unless its *N*−1 or/and *N*+1 can be ≥ 25% factored).

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large Athena primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all Athena primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

The Athena primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36 which are proven primes with ''N''−1 or ''N''+1 proving includes the Athena primes whose ''N''−1 or ''N''+1 is trivially fully factored:

* the 3176th Athena prime in base 13, 81010&lt;sub&gt;415&lt;/sub&gt;1, which equals 17746×13&lt;sup&gt;416&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000003590431555, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000003590431556&amp;open=ecm
* the 3177th Athena prime in base 13, 8110&lt;sub&gt;435&lt;/sub&gt;1, which equals 1366×13&lt;sup&gt;436&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000002373259109, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000002373259124&amp;open=ecm
* the 3188th Athena prime in base 13, 930&lt;sub&gt;1551&lt;/sub&gt;1, which equals 120×13&lt;sup&gt;1552&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000765961452, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000765961453&amp;open=ecm
* the 3191st Athena prime in base 13, 390&lt;sub&gt;6266&lt;/sub&gt;1, which equals 48×13&lt;sup&gt;6267&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000765961441, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000765961451&amp;open=ecm
* the 649th Athena prime in base 14, 34D&lt;sub&gt;708&lt;/sub&gt;, which equals 47×14&lt;sup&gt;708&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000001540144903, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000001540144907&amp;open=ecm
* the 650th Athena prime in base 14, 4D&lt;sub&gt;19698&lt;/sub&gt;, which equals 5×14&lt;sup&gt;19698&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000884560233, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000000884560625&amp;open=ecm
* the 2335th Athena prime in base 16, 88F&lt;sub&gt;545&lt;/sub&gt;, which equals 137×16&lt;sup&gt;545&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000413679658, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000000413877337&amp;open=ecm
* the 10317th Athena prime in base 17, 5A70&lt;sub&gt;274&lt;/sub&gt;1, which equals 1622×17&lt;sup&gt;275&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000003782940709, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000003782941930&amp;open=ecm
* the 10359th Athena prime in base 17, 9D0&lt;sub&gt;1067&lt;/sub&gt;1, which equals 166×17&lt;sup&gt;1068&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000765961369, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000765961370&amp;open=ecm
* the 10370th Athena prime in base 17, A0&lt;sub&gt;1355&lt;/sub&gt;1, which equals 10×17&lt;sup&gt;1356&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000034167087, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000271866825&amp;open=ecm
* the 10386th Athena prime in base 17, 530&lt;sub&gt;4867&lt;/sub&gt;1, which equals 88×17&lt;sup&gt;4868&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000762660735, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000762660737&amp;open=ecm
* the 10408th Athena prime in base 17, 570&lt;sub&gt;51310&lt;/sub&gt;1, which equals 92×17&lt;sup&gt;51311&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000765961389, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000785469616&amp;open=ecm
* the 10412th Athena prime in base 17, 970&lt;sub&gt;166047&lt;/sub&gt;1, which equals 160×17&lt;sup&gt;166048&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, but it has no helper file in ''factordb'' since it is too large (&gt;10&lt;sup&gt;199999&lt;/sup&gt;) to be PRP-tested in ''factordb'', for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000890817312&amp;open=ecm
* the 10413th Athena prime in base 17, F70&lt;sub&gt;186767&lt;/sub&gt;1, which equals 262×17&lt;sup&gt;186768&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, but it has no helper file in ''factordb'' since it is too large (&gt;10&lt;sup&gt;199999&lt;/sup&gt;) to be PRP-tested in ''factordb'', for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000890817317&amp;open=ecm
* the 3310th Athena prime in base 20, JCJ&lt;sub&gt;629&lt;/sub&gt;, which equals 393×20&lt;sup&gt;629&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000001559454258, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000001559454271&amp;open=ecm
* the 13373rd Athena prime in base 21, 5D0&lt;sub&gt;19848&lt;/sub&gt;1, which equals 118×21&lt;sup&gt;19849&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000777265872, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000785469310&amp;open=ecm
* the 3408th Athena prime in base 24, 88N&lt;sub&gt;5951&lt;/sub&gt;, which equals 201×24&lt;sup&gt;5951&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000003593275880, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000003593373246&amp;open=ecm
* the 25509th Athena prime in base 28, EB0&lt;sub&gt;405&lt;/sub&gt;1, which equals 403×28&lt;sup&gt;406&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000001534442374, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000001534442380&amp;open=ecm
* the 2616th Athena prime in base 30, C0&lt;sub&gt;1022&lt;/sub&gt;1, which equals 12×30&lt;sup&gt;1023&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000785448736, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000785448737&amp;open=ecm
* the 2619th Athena prime in base 30, OT&lt;sub&gt;34205&lt;/sub&gt;, which equals 25×30&lt;sup&gt;34205&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000800812865, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000000819405041&amp;open=ecm
* the 35237th Athena prime in base 36, P8Z&lt;sub&gt;390&lt;/sub&gt;, which equals 909×36&lt;sup&gt;390&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000764100228, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000000764100231&amp;open=ecm

and the Athena primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36 whose ''N''−1 or ''N''+1 is ≥ 1/3 factored: (''R''&lt;sub&gt;''n''&lt;/sub&gt;(''b'') means the [[:w:Repunit|repunit]] in base ''b'' with length ''n''), i.e. ''R''&lt;sub&gt;''n''&lt;/sub&gt;(''b'') = (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1), &quot;''S''&lt;sub&gt;''n''&lt;/sub&gt;(''b'')&quot; means ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)

* the 3168th Athena prime in base 13, 9&lt;sub&gt;308&lt;/sub&gt;1, ''N''−1 is 117×''R''&lt;sub&gt;308&lt;/sub&gt;(13), thus factor ''N''−1 is equivalent to factor the Cunningham number 13&lt;sup&gt;308&lt;/sup&gt;−1, and for the algebraic factors of 13&lt;sup&gt;308&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&amp;Exp=308&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 13&lt;sup&gt;308&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=308&amp;c0=-&amp;EN=&amp;LM=
* the 3179th Athena prime in base 13, B&lt;sub&gt;563&lt;/sub&gt;C, ''N''−1 is 11×''R''&lt;sub&gt;564&lt;/sub&gt;(13), thus factor ''N''−1 is equivalent to factor the Cunningham number 13&lt;sup&gt;564&lt;/sup&gt;−1, and for the algebraic factors of 13&lt;sup&gt;564&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&amp;Exp=564&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 13&lt;sup&gt;564&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=564&amp;c0=-&amp;EN=&amp;LM=
* the 3180th Athena prime in base 13, 1B&lt;sub&gt;576&lt;/sub&gt;, ''N''−1 is 23×''R''&lt;sub&gt;576&lt;/sub&gt;(13), thus factor ''N''−1 is equivalent to factor the Cunningham number 13&lt;sup&gt;576&lt;/sup&gt;−1, and for the algebraic factors of 13&lt;sup&gt;576&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&amp;Exp=576&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 13&lt;sup&gt;576&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=576&amp;c0=-&amp;EN=&amp;LM=
* the 10320th Athena prime in base 17, 9&lt;sub&gt;292&lt;/sub&gt;1, ''N''−1 is 153×''R''&lt;sub&gt;292&lt;/sub&gt;(17), thus factor ''N''−1 is equivalent to factor the Cunningham number 17&lt;sup&gt;292&lt;/sup&gt;−1, and for the algebraic factors of 17&lt;sup&gt;292&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=17&amp;Exp=292&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 17&lt;sup&gt;292&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=17&amp;Exp=292&amp;c0=-&amp;EN=&amp;LM=
* the 13304th Athena prime in base 21, 7&lt;sub&gt;230&lt;/sub&gt;1, ''N''−1 is 147×''R''&lt;sub&gt;230&lt;/sub&gt;(21), thus factor ''N''−1 is equivalent to factor the Cunningham number 21&lt;sup&gt;230&lt;/sup&gt;−1, and for the algebraic factors of 21&lt;sup&gt;230&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=21&amp;Exp=230&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 21&lt;sup&gt;230&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=21&amp;Exp=230&amp;c0=-&amp;EN=&amp;LM=
* the 13355th Athena prime in base 21, 3&lt;sub&gt;1063&lt;/sub&gt;2, ''N''+1 is 3×''R''&lt;sub&gt;1064&lt;/sub&gt;(21), thus factor ''N''−1 is equivalent to factor the Cunningham number 21&lt;sup&gt;1064&lt;/sup&gt;−1, and for the algebraic factors of 21&lt;sup&gt;1064&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=21&amp;Exp=1064&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 21&lt;sup&gt;1064&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=21&amp;Exp=1064&amp;c0=-&amp;EN=&amp;LM=
* the 25199th Athena prime in base 26, 9K&lt;sub&gt;343&lt;/sub&gt;AP, ''N''+1 is 6370×''R''&lt;sub&gt;344&lt;/sub&gt;(26), thus factor ''N''+1 is equivalent to factor the Cunningham number 26&lt;sup&gt;344&lt;/sup&gt;−1, and for the algebraic factors of 26&lt;sup&gt;344&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&amp;Exp=344&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 26&lt;sup&gt;344&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&amp;Exp=344&amp;c0=-&amp;EN=&amp;LM=
* the 25200th Athena prime in base 26, 8&lt;sub&gt;354&lt;/sub&gt;1, ''N''−1 is 208×''R''&lt;sub&gt;354&lt;/sub&gt;(26), thus factor ''N''−1 is equivalent to factor the Cunningham number 26&lt;sup&gt;354&lt;/sup&gt;−1, and for the algebraic factors of 26&lt;sup&gt;354&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&amp;Exp=354&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 26&lt;sup&gt;354&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&amp;Exp=354&amp;c0=-&amp;EN=&amp;LM=

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 Athena prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 Athena primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 Athena primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 Athena primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}

Base 6: 11 Athena primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 Athena primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331}

Base 8: 75 Athena primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}

Base 9: 151 Athena primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011}

Base 10: 77 Athena primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

Base 11: 1068 Athena (probable) primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena (probable) primes base 11]

Base 12: 106 Athena primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 Athena (probable) primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena (probable) primes base 13]

Base 14: 650 Athena primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena primes base 14]

Base 15: 1284 Athena primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena primes base 15]

Base 16: 2347 Athena (probable) primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena (probable) primes base 16]

Base 17: 10415 known Athena (probable) primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of known Athena (probable) primes base 17]

Base 18: 549 Athena primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena primes base 18]

Base 19: 31417 known Athena (probable) primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of known Athena (probable) primes base 19]

Base 20: 3314 Athena primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena primes base 20]

Base 21: 13386 known Athena (probable) primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of known Athena (probable) primes base 21]

Base 22: 8003 Athena (probable) primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena (probable) primes base 22]

Base 23: 65178 known Athena (probable) primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of known Athena (probable) primes base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for Athena problem base 23]

Base 24: 3409 Athena primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena primes base 24]

Base 25: 133639 known Athena (probable) primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of known Athena (probable) primes base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for Athena problem base 25]

Base 26: 25256 known Athena (probable) primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of known Athena (probable) primes base 26]

Base 27: 102852 known Athena (probable) primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of known Athena (probable) primes base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for Athena problem base 27]

Base 28: 25528 known Athena (probable) primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of known Athena (probable) primes base 28]

Base 29: 355242 known Athena (probable) primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of known Athena (probable) primes base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for Athena problem base 29]

Base 30: 2619 Athena (probable) primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena (probable) primes base 30]

Base 31: 569323 known Athena (probable) primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of known Athena (probable) primes base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for Athena problem base 31]

Base 32: 168882 known Athena (probable) primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of known Athena (probable) primes base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for Athena problem base 32]

Base 33: 280012 known Athena (probable) primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of known Athena (probable) primes base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for Athena problem base 33]

Base 34: 184785 known Athena (probable) primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of known Athena (probable) primes base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for Athena problem base 34]

Base 35: 720002 known Athena (probable) primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of known Athena (probable) primes base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for Athena problem base 35]

Base 36: 35286 known Athena (probable) primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of known Athena (probable) primes base 36]

== Condensed table for bases 2 ≤ ''b'' ≤ 36 ==

{|class=&quot;wikitable&quot;
||''b''||number of Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||base-''b'' form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' (write &quot;''d''&lt;sub&gt;''n''&lt;/sub&gt;&quot; if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' in decimal||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||''factordb'' entry of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of unsolved families in the Athena problem in base ''b'' (all of these left families are linear families)||searching limit of length for the unsolved families in the Athena problem in base ''b'' (if there are different searching limits for the unsolved families in the Athena problem in base ''b'', choose the lowest searching limit)||
|-
||2||1||11||2||1||3||http://factordb.com/index.php?id=3&amp;open=ecm||http://factordb.com/index.php?showid=3&amp;base=2||0||–||
|-
||3||3||111&lt;br&gt;21&lt;br&gt;12||3&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;1&lt;br&gt;1||13&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=13&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=3&lt;nowiki/&gt;||0||–||
|-
||4||5||221&lt;br&gt;31&lt;br&gt;23&lt;br&gt;13&lt;br&gt;11||3&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;1&lt;br&gt;1||41&lt;br&gt;13&lt;br&gt;11&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=41&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=41&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=4&lt;nowiki/&gt;||0||–||
|-
||5||22||10&lt;sub&gt;93&lt;/sub&gt;13&lt;br&gt;300031&lt;br&gt;44441&lt;br&gt;33331&lt;br&gt;33001&lt;br&gt;30301&lt;br&gt;14444&lt;br&gt;10103&lt;br&gt;3101&lt;br&gt;414||96&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4&lt;br&gt;3||67&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;3&lt;br&gt;3&lt;br&gt;3||5&lt;sup&gt;95&lt;/sup&gt;+8&lt;br&gt;9391&lt;br&gt;3121&lt;br&gt;2341&lt;br&gt;2251&lt;br&gt;1951&lt;br&gt;1249&lt;br&gt;653&lt;br&gt;401&lt;br&gt;109||http://factordb.com/index.php?id=1100000000034686071&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9391&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=3121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2341&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2251&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1951&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1249&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=653&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=401&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=109&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000034686071&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=9391&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=3121&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2341&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2251&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1951&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1249&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=653&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=401&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=109&amp;base=5&lt;nowiki/&gt;||0||–||
|-
||6||11||40041&lt;br&gt;4441&lt;br&gt;4401&lt;br&gt;51&lt;br&gt;45&lt;br&gt;35&lt;br&gt;31&lt;br&gt;25&lt;br&gt;21&lt;br&gt;15||5&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||5209&lt;br&gt;1033&lt;br&gt;1009&lt;br&gt;31&lt;br&gt;29&lt;br&gt;23&lt;br&gt;19&lt;br&gt;17&lt;br&gt;13&lt;br&gt;11||http://factordb.com/index.php?id=5209&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1033&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1009&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=31&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=29&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=23&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=19&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=17&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=5209&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1033&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1009&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=31&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=29&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=23&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=19&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=17&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=6&lt;nowiki/&gt;||0||–||
|-
||7||71||3&lt;sub&gt;16&lt;/sub&gt;1&lt;br&gt;510&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;3&lt;sub&gt;6&lt;/sub&gt;01&lt;br&gt;1100021&lt;br&gt;531101&lt;br&gt;351101&lt;br&gt;300053&lt;br&gt;150001&lt;br&gt;100121&lt;br&gt;40054||17&lt;br&gt;10&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;5||15&lt;br&gt;9&lt;br&gt;7&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2&lt;br&gt;36×7&lt;sup&gt;8&lt;/sup&gt;+1&lt;br&gt;(7&lt;sup&gt;8&lt;/sup&gt;−47)/2&lt;br&gt;134471&lt;br&gt;91631&lt;br&gt;62819&lt;br&gt;50459&lt;br&gt;28813&lt;br&gt;16871&lt;br&gt;9643||http://factordb.com/index.php?id=116315256993601&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=207532837&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2882377&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=134471&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=91631&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=62819&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=50459&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=28813&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=16871&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9643&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=116315256993601&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=207532837&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=2882377&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=134471&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=91631&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=62819&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=50459&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=28813&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=16871&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=9643&amp;base=7&lt;nowiki/&gt;||0||–||
|-
||8||75||4&lt;sub&gt;220&lt;/sub&gt;7&lt;br&gt;5&lt;sub&gt;13&lt;/sub&gt;25&lt;br&gt;7&lt;sub&gt;12&lt;/sub&gt;1&lt;br&gt;77774&lt;sub&gt;6&lt;/sub&gt;1&lt;br&gt;74&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;4&lt;sub&gt;8&lt;/sub&gt;1&lt;br&gt;5&lt;sub&gt;5&lt;/sub&gt;025&lt;br&gt;5550525&lt;br&gt;5500525&lt;br&gt;4&lt;sub&gt;5&lt;/sub&gt;77||221&lt;br&gt;15&lt;br&gt;13&lt;br&gt;11&lt;br&gt;9&lt;br&gt;9&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||200&lt;br&gt;14&lt;br&gt;12&lt;br&gt;10&lt;br&gt;9&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7&lt;br&gt;(5×8&lt;sup&gt;15&lt;/sup&gt;−173)/7&lt;br&gt;8&lt;sup&gt;13&lt;/sup&gt;−7&lt;br&gt;(28669×8&lt;sup&gt;7&lt;/sup&gt;−25)/7&lt;br&gt;(53×8&lt;sup&gt;8&lt;/sup&gt;−25)/7&lt;br&gt;(4×8&lt;sup&gt;9&lt;/sup&gt;−25)/7&lt;br&gt;(5×8&lt;sup&gt;8&lt;/sup&gt;−2413)/7&lt;br&gt;1495381&lt;br&gt;1474901&lt;br&gt;(4×8&lt;sup&gt;7&lt;/sup&gt;+185)/7||http://factordb.com/index.php?id=1100000000416605822&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=25131694349141&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=549755813881&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=8589035809&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=127027489&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=76695841&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11983381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1495381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1474901&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1198399&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000416605822&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=25131694349141&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=549755813881&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=8589035809&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=127027489&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=76695841&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=11983381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1495381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1474901&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1198399&amp;base=8&lt;nowiki/&gt;||0||–||
|-
||9||151||30&lt;sub&gt;1158&lt;/sub&gt;11&lt;br&gt;27&lt;sub&gt;686&lt;/sub&gt;07&lt;br&gt;76&lt;sub&gt;329&lt;/sub&gt;2&lt;br&gt;561&lt;sub&gt;36&lt;/sub&gt;&lt;br&gt;10&lt;sub&gt;25&lt;/sub&gt;57&lt;br&gt;30&lt;sub&gt;20&lt;/sub&gt;51&lt;br&gt;8&lt;sub&gt;19&lt;/sub&gt;335&lt;br&gt;727&lt;sub&gt;15&lt;/sub&gt;07&lt;br&gt;51&lt;sub&gt;13&lt;/sub&gt;61&lt;br&gt;10&lt;sub&gt;11&lt;/sub&gt;507||1161&lt;br&gt;689&lt;br&gt;331&lt;br&gt;38&lt;br&gt;28&lt;br&gt;23&lt;br&gt;22&lt;br&gt;19&lt;br&gt;16&lt;br&gt;15||1108&lt;br&gt;657&lt;br&gt;316&lt;br&gt;37&lt;br&gt;26&lt;br&gt;22&lt;br&gt;21&lt;br&gt;19&lt;br&gt;16&lt;br&gt;14||3×9&lt;sup&gt;1160&lt;/sup&gt;+10&lt;br&gt;(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8&lt;br&gt;(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4&lt;br&gt;(409×9&lt;sup&gt;36&lt;/sup&gt;−1)/8&lt;br&gt;9&lt;sup&gt;27&lt;/sup&gt;+52&lt;br&gt;3×9&lt;sup&gt;22&lt;/sup&gt;+46&lt;br&gt;9&lt;sup&gt;22&lt;/sup&gt;−454&lt;br&gt;(527×9&lt;sup&gt;17&lt;/sup&gt;−511)/8&lt;br&gt;(41×9&lt;sup&gt;15&lt;/sup&gt;+359)/8&lt;br&gt;9&lt;sup&gt;14&lt;/sup&gt;+412||http://factordb.com/index.php?id=1100000002376318423&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002495467486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002359003642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000001554010824&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002512830927&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000000032261811&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000002495736583&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1100000003446800389&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1055192051985121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=22876792455373&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002376318423&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495467486&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002359003642&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000001554010824&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002512830927&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000000032261811&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000002495736583&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1100000003446800389&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=1055192051985121&amp;base=9&lt;br&gt;http://factordb.com/index.php?showid=22876792455373&amp;base=9&lt;nowiki/&gt;||0||–||
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||23||65178~65265||B0&lt;sub&gt;93046&lt;/sub&gt;FB&lt;br&gt;L&lt;sub&gt;86444&lt;/sub&gt;D&lt;br&gt;AJ&lt;sub&gt;81065&lt;/sub&gt;4&lt;br&gt;20&lt;sub&gt;73560&lt;/sub&gt;98&lt;br&gt;J&lt;sub&gt;68217&lt;/sub&gt;G4&lt;br&gt;D70&lt;sub&gt;66770&lt;/sub&gt;B&lt;br&gt;5F&lt;sub&gt;62340&lt;/sub&gt;6&lt;br&gt;A7M7&lt;sub&gt;61532&lt;/sub&gt;&lt;br&gt;B30&lt;sub&gt;61136&lt;/sub&gt;5&lt;br&gt;EJ&lt;sub&gt;52169&lt;/sub&gt;||93049&lt;br&gt;86445&lt;br&gt;81067&lt;br&gt;73563&lt;br&gt;68219&lt;br&gt;66773&lt;br&gt;62342&lt;br&gt;61535&lt;br&gt;61139&lt;br&gt;52170||126708&lt;br&gt;117715&lt;br&gt;110391&lt;br&gt;100172&lt;br&gt;92896&lt;br&gt;90927&lt;br&gt;84893&lt;br&gt;83794&lt;br&gt;83255&lt;br&gt;71042||11×23&lt;sup&gt;93048&lt;/sup&gt;+356&lt;br&gt;(21×23&lt;sup&gt;86445&lt;/sup&gt;−197)/22&lt;br&gt;(239×23&lt;sup&gt;81066&lt;/sup&gt;−349)/22&lt;br&gt;2×23&lt;sup&gt;73562&lt;/sup&gt;+215&lt;br&gt;(19×23&lt;sup&gt;68219&lt;/sup&gt;−1867)/22&lt;br&gt;306×23&lt;sup&gt;66771&lt;/sup&gt;+11&lt;br&gt;(125×23&lt;sup&gt;62341&lt;/sup&gt;−213)/22&lt;br&gt;(120413×23&lt;sup&gt;61532&lt;/sup&gt;−7)/22&lt;br&gt;256×23&lt;sup&gt;61137&lt;/sup&gt;+5&lt;br&gt;(327×23&lt;sup&gt;52169&lt;/sup&gt;−19)/22||http://factordb.com/index.php?id=1100000004691540361&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691546739&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691548070&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691548569&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691549462&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691549803&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691551005&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691556967&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691557254&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004691557548&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000004691540361&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691546739&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691548070&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691548569&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691549462&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691549803&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691551005&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691556967&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691557254&amp;base=23&lt;br&gt;http://factordb.com/index.php?showid=1100000004691557548&amp;base=23&lt;nowiki/&gt;||87||100000||
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||24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN&lt;br&gt;88N&lt;sub&gt;5951&lt;/sub&gt;&lt;br&gt;A0&lt;sub&gt;2951&lt;/sub&gt;8ID&lt;br&gt;D&lt;sub&gt;2698&lt;/sub&gt;LD&lt;br&gt;N&lt;sub&gt;2644&lt;/sub&gt;LLN&lt;br&gt;BC0&lt;sub&gt;331&lt;/sub&gt;B&lt;br&gt;20&lt;sub&gt;313&lt;/sub&gt;7&lt;br&gt;C7&lt;sub&gt;298&lt;/sub&gt;&lt;br&gt;D0&lt;sub&gt;259&lt;/sub&gt;KKD&lt;br&gt;I0&lt;sub&gt;241&lt;/sub&gt;I5||8134&lt;br&gt;5953&lt;br&gt;2955&lt;br&gt;2700&lt;br&gt;2647&lt;br&gt;334&lt;br&gt;315&lt;br&gt;299&lt;br&gt;263&lt;br&gt;244||11227&lt;br&gt;8216&lt;br&gt;4079&lt;br&gt;3727&lt;br&gt;3654&lt;br&gt;461&lt;br&gt;434&lt;br&gt;413&lt;br&gt;363&lt;br&gt;337||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49&lt;br&gt;201×24&lt;sup&gt;5951&lt;/sup&gt;−1&lt;br&gt;10×24&lt;sup&gt;2954&lt;/sup&gt;+5053&lt;br&gt;(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23&lt;br&gt;24&lt;sup&gt;2647&lt;/sup&gt;−1201&lt;br&gt;276×24&lt;sup&gt;332&lt;/sup&gt;+11&lt;br&gt;2×24&lt;sup&gt;314&lt;/sup&gt;+7&lt;br&gt;(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23&lt;br&gt;13×24&lt;sup&gt;262&lt;/sup&gt;+12013&lt;br&gt;18×24&lt;sup&gt;243&lt;/sup&gt;+437||http://factordb.com/index.php?id=1100000003593391606&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593275880&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269654&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593269876&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270089&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633359842&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002355610241&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002326181235&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003593270725&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000002633360037&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000003593391606&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593275880&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269654&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593269876&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270089&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633359842&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002355610241&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002326181235&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000003593270725&amp;base=24&lt;br&gt;http://factordb.com/index.php?showid=1100000002633360037&amp;base=24&lt;nowiki/&gt;||0||–||
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|-
||35||720002~720062||N0N&lt;sub&gt;99971&lt;/sub&gt;9&lt;br&gt;V0&lt;sub&gt;83669&lt;/sub&gt;E73&lt;br&gt;N&lt;sub&gt;81563&lt;/sub&gt;K7N&lt;br&gt;BJ0&lt;sub&gt;81279&lt;/sub&gt;N&lt;br&gt;J0&lt;sub&gt;80062&lt;/sub&gt;FUH&lt;br&gt;43V&lt;sub&gt;79754&lt;/sub&gt;&lt;br&gt;9&lt;sub&gt;76600&lt;/sub&gt;K3&lt;br&gt;LB&lt;sub&gt;71366&lt;/sub&gt;PB&lt;br&gt;Q&lt;sub&gt;64150&lt;/sub&gt;H&lt;br&gt;50&lt;sub&gt;63397&lt;/sub&gt;5R||99974&lt;br&gt;83673&lt;br&gt;81566&lt;br&gt;81282&lt;br&gt;80066&lt;br&gt;79756&lt;br&gt;76602&lt;br&gt;71369&lt;br&gt;64151&lt;br&gt;63400||154367&lt;br&gt;129197&lt;br&gt;125944&lt;br&gt;125505&lt;br&gt;123628&lt;br&gt;123148&lt;br&gt;118279&lt;br&gt;110199&lt;br&gt;99054&lt;br&gt;97894||(27393×35&lt;sup&gt;99972&lt;/sup&gt;−499)/34&lt;br&gt;31×35&lt;sup&gt;83672&lt;/sup&gt;+17398&lt;br&gt;(23×35&lt;sup&gt;81566&lt;/sup&gt;−144013)/34&lt;br&gt;404×35&lt;sup&gt;81280&lt;/sup&gt;+23&lt;br&gt;19×35&lt;sup&gt;80065&lt;/sup&gt;+19442&lt;br&gt;(4893×35&lt;sup&gt;79754&lt;/sup&gt;−31)/34&lt;br&gt;(9×35&lt;sup&gt;76602&lt;/sup&gt;+12877)/34&lt;br&gt;(725×35&lt;sup&gt;71368&lt;/sup&gt;+16649)/34&lt;br&gt;(13×35&lt;sup&gt;64151&lt;/sup&gt;−166)/17&lt;br&gt;5×35&lt;sup&gt;63399&lt;/sup&gt;+202||http://factordb.com/index.php?id=1100000008248342445&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248353306&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248375642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248397018&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248412468&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248418540&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248423670&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192119974&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192126630&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192129294&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000008248342445&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248353306&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248375642&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248397018&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248412468&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248418540&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248423670&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192119974&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192126630&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192129294&amp;base=35&lt;nowiki/&gt;||60||100000||
|-
||36||35286~35290||P&lt;sub&gt;81993&lt;/sub&gt;SZ&lt;br&gt;S0&lt;sub&gt;75007&lt;/sub&gt;8H&lt;br&gt;7K&lt;sub&gt;26567&lt;/sub&gt;Z&lt;br&gt;J&lt;sub&gt;10117&lt;/sub&gt;LJ&lt;br&gt;VL0&lt;sub&gt;7258&lt;/sub&gt;J&lt;br&gt;EO0&lt;sub&gt;6177&lt;/sub&gt;V&lt;br&gt;FZ&lt;sub&gt;5777&lt;/sub&gt;3P&lt;br&gt;T09&lt;sub&gt;4618&lt;/sub&gt;1&lt;br&gt;RY&lt;sub&gt;4562&lt;/sub&gt;H&lt;br&gt;OZ&lt;sub&gt;3932&lt;/sub&gt;AZ||81995&lt;br&gt;75010&lt;br&gt;26569&lt;br&gt;10119&lt;br&gt;7261&lt;br&gt;6180&lt;br&gt;5780&lt;br&gt;4621&lt;br&gt;4564&lt;br&gt;3935||127609&lt;br&gt;116739&lt;br&gt;41349&lt;br&gt;15748&lt;br&gt;11301&lt;br&gt;9618&lt;br&gt;8996&lt;br&gt;7192&lt;br&gt;7103&lt;br&gt;6124||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7&lt;br&gt;28×36&lt;sup&gt;75009&lt;/sup&gt;+305&lt;br&gt;(53×36&lt;sup&gt;26568&lt;/sup&gt;+101)/7&lt;br&gt;(19×36&lt;sup&gt;10119&lt;/sup&gt;+2501)/35&lt;br&gt;1137×36&lt;sup&gt;7259&lt;/sup&gt;+19&lt;br&gt;528×36&lt;sup&gt;6178&lt;/sup&gt;+31&lt;br&gt;16×36&lt;sup&gt;5779&lt;/sup&gt;−1163&lt;br&gt;(36549×36&lt;sup&gt;4619&lt;/sup&gt;−289)/35&lt;br&gt;(979×36&lt;sup&gt;4563&lt;/sup&gt;−629)/35&lt;br&gt;25×36&lt;sup&gt;3934&lt;/sup&gt;−901||http://factordb.com/index.php?id=1100000002394962083&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004020085177&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003896952461&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362491&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362489&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362488&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362487&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362485&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000000840634476&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002394962083&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000004020085177&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003896952461&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362491&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362489&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362488&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362487&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362486&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362485&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000000840634476&amp;base=36&lt;nowiki/&gt;||4||200000||
|}

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the Athena primes, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1 (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s), this set is called '''Athena set''', and the prime numbers in this set are called '''Athena primes'''.

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the Athena set in every base ''b'' must be a [[:w:Finite set|finite set]], and every base ''b'' ≥ 2 has only finitely many Athena primes, e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), the Athena set has exactly 77 [[:w:Element of a set|element]]s (they are exactly the Athena primes in decimal (base ''b'' = 10)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}.

Determining the set of the minimal elements of a arbitrary set of strings under the subsequence ordering is in general [[:w:List of unsolved problems in mathematics|unsolvable]], and can be difficult even when this set is relatively simple (such as the base ''b'' representations of the prime numbers &gt; ''b'', whose set is exactly the Athena set in base ''b'').

Although the set ''M''(''S'') of minimal strings is necessarily [[:w:Finite set|finite]], determining it explicitly for a given ''S'' can be a difficult computational problem. We use some [[:w:Number theory|numbertheoretic]] [[:w:Heuristic argument|heuristic]]s to [[:w:Computing|compute]] ''M''(''L''&lt;sub&gt;''b''&lt;/sub&gt;) (i.e. to compute the Athena set in base ''b''), where ''L''&lt;sub&gt;''b''&lt;/sub&gt; is the [[:w:Formal language|language]] of [[:w:Radix|base]]-''b'' representations of the [[:w:Prime number|prime number]]s which are [[:w:Greater than|&gt;]] ''b'', for 2 ≤ ''b'' ≤ 36.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000. (The chance that an unproven probable prime in these sets is in fact composite is less than 10&lt;sup&gt;−2000&lt;/sup&gt;, see https://t5k.org/notes/prp_prob.html)

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of Athena primes in base ''b''&quot; and &quot;length of the largest Athena prime in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the Athena set in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem (i.e. to compute the Athena set), we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist) (in fact, it is known that exactly what bases 2 ≤ ''b'' ≤ 1024 have the families listed in the table below as unsolved families, all of these families in all bases 2 ≤ ''b'' ≤ 1024 have been searched to length ≥ 10000 (for the family (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1), bases 2 ≤ ''b'' ≤ 1048576, searched to length ≥ 5000)): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)):

'''Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?'''

An [[:w:Algorithm|algorithm]] to solve this problem, for example, would allow us to decide if there are any additional [[:w:Fermat prime|Fermat prime]]s other than the known ones (corresponding to ''n'' = 0, 1, 2, 3, 4). To see this, take ''b'' = 2, ''x'' = 1, ''y'' = 0, and ''z'' = 0&lt;sub&gt;16&lt;/sub&gt;1. Since if 2&lt;sup&gt;''n''&lt;/sup&gt;+1 is prime then ''n'' must be a [[:w:Power of 2|power of two]], a prime of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' in base ''b'' must be a new Fermat prime. Besides, it would allow us to decide if there are infinitely many [[:w:Mersenne prime|Mersenne prime]]s (of the form 2&lt;sup&gt;''p''&lt;/sup&gt;−1 with prime ''p''). To see this, take ''b'' = 2, ''x'' = ''𝜆'' (the [[:w:Empty string|empty string]]), ''y'' = 1, and ''z'' = 1&lt;sub&gt;''n''+1&lt;/sub&gt;, where ''n'' is the exponent of the Mersenne prime which we want to know whether it is the largest Mersenne prime or not. Since if 2&lt;sup&gt;''n''&lt;/sup&gt;−1 is prime then ''n'' must be a [[:w:Prime number|prime]], a prime of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' in base ''b'' must be a new Mersenne prime. Also, it would allow us to decide whether 78557 is the smallest [[:w:Sierpinski number|Sierpinski number]] (i.e. odd numbers ''k'' such that ''k''×2&lt;sup&gt;''n''&lt;/sup&gt;+1 is composite for all ''n'' ≥ 1) and whether 509203 is the smallest [[:w:Riesel number|Riesel number]] (i.e. odd numbers ''k'' such that ''k''×2&lt;sup&gt;*n*&lt;/sup&gt;−1 is composite for all ''n'' ≥ 1), etc.

'''Athena conjecture (which is very important for the Athena problem): If family ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' (with fixed strings ''x'', ''z'' (may be empty), fixed digit ''y'', and variable ''n'') in base ''b'' (with fixed ''b'' ≥ 2) (''x'' does not start with the digit 0, ''z'' ends with a digit which coprime to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty) cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then family ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' in base ''b'' contains infinitely many primes (this is equivalent to: If form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) (with fixed integers ''a'' ≥ 1, ''b'' ≥ 2, ''c'' ≠ 0 (with ''gcd''(''a'',''c'') = 1 and ''gcd''(''b'',''c'') = 1), and variable ''n'') cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) contains infinitely many primes).'''

(in fact, the Athena conjecture is equivalent to the conjecture (to prove this, by change the base (''b'') to a power of ''b'' which is larger than the largest prime in a given family (in base ''b'') which only contains finitely many primes): If family ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' (with fixed strings ''x'', ''z'' (may be empty), fixed digit ''y'', and variable ''n'') in base ''b'' (with fixed ''b'' ≥ 2) (''x'' does not start with the digit 0, ''z'' ends with a digit which coprime to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty) cannot be proven to only contain composites (by covering congruence, algebraic factorization, or combine of them), then family ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' in base ''b'' contains at least one prime (this is equivalent to: If form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) (with fixed integers ''a'' ≥ 1, ''b'' ≥ 2, ''c'' ≠ 0 (with ''gcd''(''a'',''c'') = 1 and ''gcd''(''b'',''c'') = 1), and variable ''n'') cannot be proven to only contain composites (by covering congruence, algebraic factorization, or combine of them), then form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) contains at least one prime), like the [[:w:Bunyakovsky conjecture|Bunyakovsky conjecture]] and the [[:w:Dickson's conjecture|Dickson's conjecture]] and the [[:w:Schinzel's hypothesis H|Schinzel's hypothesis ''H'']], if such ''n'' always exists, then there must be always infinitely many such ''n'', to prove this, add another polynomial for the cases of the Dickson's conjecture and the Schinzel's hypothesis ''H'', also, change the polynomial (e.g. change ''n'' to ''r''×''n'' or ''n''&lt;sup&gt;''r''&lt;/sup&gt; for all integers ''r'' &gt; 1) for the cases of the Bunyakovsky conjecture and the Schinzel's hypothesis ''H'')

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 11 family 2{5}: Always divisible by 2 or 3
* The base 14 family B{0}1: Always divisible by 3 or 5
* The base 13 family 95{0}3: Always divisible by 5, 7, or 17
* The base 16 family {4}D: Always divisible by 3, 7, or 13
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family {1}: Can be written as (9&lt;sup&gt;''n''&lt;/sup&gt;−1)/8 and can be factored as (3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 8
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family 1{5}: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−1)/3 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1) / 3
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 27 family 7{Q}: Can be written as 8×27&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (4×9&lt;sup&gt;''n''&lt;/sup&gt;+2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 27 family 9{G}: Can be written as (125×27&lt;sup&gt;''n''&lt;/sup&gt;−8)/13 and can be factored as (5×3&lt;sup&gt;''n''&lt;/sup&gt;−2) × (25×9&lt;sup&gt;''n''&lt;/sup&gt;+10×3&lt;sup&gt;''n''&lt;/sup&gt;+4)
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 14 family {D}5: Can be written as 14&lt;sup&gt;''n''&lt;/sup&gt;−9, it is divisible by 5 if ''n'' is odd and can be factored as (14&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (14&lt;sup&gt;''n''/2&lt;/sup&gt;+3) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 17 family 7{9}: Can be written as (121×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (11×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even
* The base 24 family 3{N}: Can be written as 4×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×24&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 24 family 5{N}: Can be written as 6×24&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is even and can be factored as (12×24&lt;sup&gt;(''n''−1)/2&lt;/sup&gt;−1) × (12×24&lt;sup&gt;(''n''−1)/2&lt;/sup&gt;+1) if ''n'' is odd

If the Athena conjecture is true, then the [[:w:Sierpiński number|Sierpiński conjecture]] and [[:w:Riesel number|Riesel conjecture]] are also true, and the [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm Sierpiński conjectures] and the [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm Riesel conjectures] in all bases ''b'' are also true, and the [http://www.noprimeleftbehind.net/crus/SNOB-Sierp-conjectures.htm real Sierpiński conjectures] and the [http://www.noprimeleftbehind.net/crus/Real-Riesel-conjectures.htm real Riesel conjectures] are also true, also, if the Athena conjecture is true, then there are infinitely many primes of these forms for fixed bases ''b'' ≥ 2 and variable exponents ''n'':

* (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) for all bases ''b'' which are not [[:w:Perfect power|perfect power]]s (for this form, ''n'' must be prime) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all even bases ''b'' which are not of the form ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r'' &gt; 1 (for this form, ''n'' must be power of 2) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101)
* (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 for all odd bases ''b'' which are not of the form ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r'' &gt; 1 (for this form, ''n'' must be power of 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt)
* (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''+1) for all bases ''b'' which are neither of the form ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r'' &gt; 1 nor of the form 4×''m''&lt;sup&gt;4&lt;/sup&gt; (for this form, ''n'' must be prime) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for prime bases ''b'': https://oeis.org/A065507)
* ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) for all bases ''b'' &gt; 2 (reference of this form: https://oeis.org/A243404)
* 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' &lt; 201446503145165177, not == 1 mod 3 (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624)
* 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&amp;d=1567314217, https://oeis.org/A119591)
* 3×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all even bases ''b''
* 3×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all even bases ''b''
* 4×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' not == 1 mod 5, not == 14 mod 15, not [[:w:Fourth power|fourth power]]s
* 4×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' not == 1 mod 3, not == 4 mod 5, not [[:w:Square number|square]]s
* 5×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all even bases ''b'' &lt; 140324348, not == 1 mod 3
* 5×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all even bases ''b''
* 6×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' not == 1 mod 7, not == 34 mod 35
* 6×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' not == 1 mod 5, not == 34 mod 35, not of the form 6×''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2, 3 mod 5
* 7×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all even bases ''b''
* 7×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all even bases ''b'' &lt; 9162668342, not == 1 mod 3
* 8×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' not == 1 mod 3, not == 20 mod 21, not == 47, 83 mod 195, not == 467, 4343, 9887, 25448, 35978, 41522, 42647, 57083 mod 73815, not == 722, 83813, 206672, 239432, 322523, 1283843, 1519577, 1522553 mod 1551615, ..., not [[:w:Cube (algebra)|cube]]s
* 8×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' not == 1 mod 7, not == 20 mod 21, not == 83, 307 mod 455, not == 1266, 13593, 27292, 46353 mod 63973, ..., not [[:w:Cube (algebra)|cube]]s
* 9×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all even bases ''b'' &lt; 177744, not == 1 mod 5
* 9×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all even bases ''b'' not == 4 mod 5, not [[:w:Square number|square]]s
* 10×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' not == 1 mod 11, not == 32 mod 33 (references of this form: https://oeis.org/A088782)
* 10×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' not == 1 mod 3, not == 32 mod 33
* 11×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all even bases ''b'' not == 1 mod 3, not == 14 mod 15
* 11×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all even bases ''b'' not == 1 mod 5, not == 14 mod 15, not of the form 11×''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2, 3 mod 5
* 12×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' not == 1 mod 13, not == 142 mod 143, not == 562, 828, 900, 1166 mod 1729, not == 597, 1143 mod 1885, not == 296, 901, 1759, 3090, 4553, 5521, 5807, 6016, 6984, 7094, 7270, 7380, 7479, 8447, 8557, 8733, 8843, 9910, 10020, 10196, 10306, 11483, 11769, 12737, 14200, 15531, 16994, 18457 mod 19019, not == 563, 1433, 13212, 15097, 19848, 20718, 32497, 34382, 39133, 51782, 53667, 58418, 58452, 60337, 60883, 71067, 72952, 77737, 79622, 80168, 94267, 97022, 98583, 98907, 113552, 116307, 117868, 118192, 131967, 132513, 132837, 134398, 151252, 151798, 152122, 153683, 170537, 171083, 172968, 177753, 179638, 189822, 190368, 192253, 192287, 197038, 198923, 211572, 213568, 216323, 218208, 229987, 232853, 235608, 237493, 249272 mod 250705, ...
* 12×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' not == 1 mod 11, not == 142 mod 143, not == 307, 1143 mod 1595, not == 901, 6016, 7479, 18457 mod 19019, ...
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 for all odd bases ''b'' &lt; 201446503145165177, not == 1 mod 3 (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 for all odd bases ''b'' (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;+3 for all even bases ''b'' not divisible by 3
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;−3 for all even bases ''b'' not divisible by 3
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;+4 for all odd bases ''b'' not == 1 mod 5, not == 14 mod 15, not [[:w:Fourth power|fourth power]]s
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;−4 for all odd bases ''b'' not == 1 mod 3, not == 4 mod 5, not [[:w:Square number|square]]s
* (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139)
* (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396)
* (''b''+1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 for all bases ''b'' not == 1 mod 3 (references of this form: http://www.noprimeleftbehind.net/Williams-primes-PP.htm, http://www.bitman.name/math/table/474, https://pzktupel.de/Primetables/TableWilliams4.php)
* (''b''+1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 for all bases ''b'' (references of this form: http://www.noprimeleftbehind.net/Williams-primes-PM.htm, http://www.bitman.name/math/table/471, https://pzktupel.de/Primetables/TableWilliams3.php)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) for all bases ''b'' (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) for all bases ''b'' (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''+1) for all bases ''b'' not == 1 mod 3 (references of this form: http://www.bitman.name/math/table/801, https://pzktupel.de/Primetables/TableWilliams8.php, https://oeis.org/A346149, https://oeis.org/A346154)
* ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''+1) for all bases ''b'' (references of this form: http://www.bitman.name/math/table/798, https://pzktupel.de/Primetables/TableWilliams7.php, https://oeis.org/A178250)

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored).

The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1 (the only case which [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] is [[:w:Triviality (mathematics)|trivially]] fully factored), when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run (since they are [https://t5k.org/glossary/xpage/OrdinaryPrime.html ordinary primes]). In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]].

To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1 we only used it to sieve the sequence ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'', but 2 may not divide (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)).

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) &gt; 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates.

For the primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for the &quot;easy&quot; bases (bases ''b'' with ≤ 150 primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; (base ''b'' = 26 has 83 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 3 unsolved families, base ''b'' = 36 has 75 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 4 unsolved families, base ''b'' = 17 has 99 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 18 unsolved families, base ''b'' = 21 has 80 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 12 unsolved families, base ''b'' = 19 has 201 known primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation.

Currently, the final goal of the Athena problem project is finding the Athena set (i.e. finding all Athena primes) and proving that this set is exactly the Athena set (i.e. proving that these are all Athena primes (including the primality proving for the probable primes)) in all bases 2 ≤ ''b'' ≤ 36, i.e. solving all families in all bases 2 ≤ ''b'' ≤ 36. Solving all (unsolved) families in all bases 2 ≤ ''b'' ≤ 36 (and proving the primality of all probable primes in the sets of all bases 2 ≤ ''b'' ≤ 36) is not possible but we aim to solve many of them (and proving the primality of many of them), at least find a ''probable'' prime for many of them (since the smallest prime in a family may be too large (&gt; 10&lt;sup&gt;25000&lt;/sup&gt;) to be proved primality, unless its *N*−1 or/and *N*+1 can be ≥ 25% factored).

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large Athena primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all Athena primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

The Athena primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36 which are proven primes with ''N''−1 or ''N''+1 proving includes the Athena primes whose ''N''−1 or ''N''+1 is trivially fully factored:

* the 3176th Athena prime in base 13, 81010&lt;sub&gt;415&lt;/sub&gt;1, which equals 17746×13&lt;sup&gt;416&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000003590431555, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000003590431556&amp;open=ecm
* the 3177th Athena prime in base 13, 8110&lt;sub&gt;435&lt;/sub&gt;1, which equals 1366×13&lt;sup&gt;436&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000002373259109, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000002373259124&amp;open=ecm
* the 3188th Athena prime in base 13, 930&lt;sub&gt;1551&lt;/sub&gt;1, which equals 120×13&lt;sup&gt;1552&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000765961452, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000765961453&amp;open=ecm
* the 3191st Athena prime in base 13, 390&lt;sub&gt;6266&lt;/sub&gt;1, which equals 48×13&lt;sup&gt;6267&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000765961441, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000765961451&amp;open=ecm
* the 649th Athena prime in base 14, 34D&lt;sub&gt;708&lt;/sub&gt;, which equals 47×14&lt;sup&gt;708&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000001540144903, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000001540144907&amp;open=ecm
* the 650th Athena prime in base 14, 4D&lt;sub&gt;19698&lt;/sub&gt;, which equals 5×14&lt;sup&gt;19698&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000884560233, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000000884560625&amp;open=ecm
* the 2335th Athena prime in base 16, 88F&lt;sub&gt;545&lt;/sub&gt;, which equals 137×16&lt;sup&gt;545&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000413679658, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000000413877337&amp;open=ecm
* the 10317th Athena prime in base 17, 5A70&lt;sub&gt;274&lt;/sub&gt;1, which equals 1622×17&lt;sup&gt;275&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000003782940709, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000003782941930&amp;open=ecm
* the 10359th Athena prime in base 17, 9D0&lt;sub&gt;1067&lt;/sub&gt;1, which equals 166×17&lt;sup&gt;1068&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000765961369, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000765961370&amp;open=ecm
* the 10370th Athena prime in base 17, A0&lt;sub&gt;1355&lt;/sub&gt;1, which equals 10×17&lt;sup&gt;1356&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000034167087, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000271866825&amp;open=ecm
* the 10386th Athena prime in base 17, 530&lt;sub&gt;4867&lt;/sub&gt;1, which equals 88×17&lt;sup&gt;4868&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000762660735, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000762660737&amp;open=ecm
* the 10408th Athena prime in base 17, 570&lt;sub&gt;51310&lt;/sub&gt;1, which equals 92×17&lt;sup&gt;51311&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000765961389, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000785469616&amp;open=ecm
* the 10412th Athena prime in base 17, 970&lt;sub&gt;166047&lt;/sub&gt;1, which equals 160×17&lt;sup&gt;166048&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, but it has no helper file in ''factordb'' since it is too large (&gt;10&lt;sup&gt;199999&lt;/sup&gt;) to be PRP-tested in ''factordb'', for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000890817312&amp;open=ecm
* the 10413th Athena prime in base 17, F70&lt;sub&gt;186767&lt;/sub&gt;1, which equals 262×17&lt;sup&gt;186768&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, but it has no helper file in ''factordb'' since it is too large (&gt;10&lt;sup&gt;199999&lt;/sup&gt;) to be PRP-tested in ''factordb'', for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000890817317&amp;open=ecm
* the 3310th Athena prime in base 20, JCJ&lt;sub&gt;629&lt;/sub&gt;, which equals 393×20&lt;sup&gt;629&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000001559454258, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000001559454271&amp;open=ecm
* the 13373rd Athena prime in base 21, 5D0&lt;sub&gt;19848&lt;/sub&gt;1, which equals 118×21&lt;sup&gt;19849&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000777265872, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000785469310&amp;open=ecm
* the 3408th Athena prime in base 24, 88N&lt;sub&gt;5951&lt;/sub&gt;, which equals 201×24&lt;sup&gt;5951&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000003593275880, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000003593373246&amp;open=ecm
* the 25509th Athena prime in base 28, EB0&lt;sub&gt;405&lt;/sub&gt;1, which equals 403×28&lt;sup&gt;406&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000001534442374, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000001534442380&amp;open=ecm
* the 2616th Athena prime in base 30, C0&lt;sub&gt;1022&lt;/sub&gt;1, which equals 12×30&lt;sup&gt;1023&lt;/sup&gt;+1, ''N''−1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000785448736, for the factorization of ''N''−1 in ''factordb'' see http://factordb.com/index.php?id=1100000000785448737&amp;open=ecm
* the 2619th Athena prime in base 30, OT&lt;sub&gt;34205&lt;/sub&gt;, which equals 25×30&lt;sup&gt;34205&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000800812865, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000000819405041&amp;open=ecm
* the 35237th Athena prime in base 36, P8Z&lt;sub&gt;390&lt;/sub&gt;, which equals 909×36&lt;sup&gt;390&lt;/sup&gt;−1, ''N''+1 is trivially fully factored, for its helper file in ''factordb'' see http://factordb.com/helper.php?id=1100000000764100228, for the factorization of ''N''+1 in ''factordb'' see http://factordb.com/index.php?id=1100000000764100231&amp;open=ecm

and the Athena primes &gt; 10&lt;sup&gt;299&lt;/sup&gt; in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36 whose ''N''−1 or ''N''+1 is ≥ 1/3 factored: (''R''&lt;sub&gt;''n''&lt;/sub&gt;(''b'') means the [[:w:Repunit|repunit]] in base ''b'' with length ''n''), i.e. ''R''&lt;sub&gt;''n''&lt;/sub&gt;(''b'') = (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1), &quot;''S''&lt;sub&gt;''n''&lt;/sub&gt;(''b'')&quot; means ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)

* the 3168th Athena prime in base 13, 9&lt;sub&gt;308&lt;/sub&gt;1, ''N''−1 is 117×''R''&lt;sub&gt;308&lt;/sub&gt;(13), thus factor ''N''−1 is equivalent to factor the Cunningham number 13&lt;sup&gt;308&lt;/sup&gt;−1, and for the algebraic factors of 13&lt;sup&gt;308&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&amp;Exp=308&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 13&lt;sup&gt;308&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=308&amp;c0=-&amp;EN=&amp;LM=
* the 3179th Athena prime in base 13, B&lt;sub&gt;563&lt;/sub&gt;C, ''N''−1 is 11×''R''&lt;sub&gt;564&lt;/sub&gt;(13), thus factor ''N''−1 is equivalent to factor the Cunningham number 13&lt;sup&gt;564&lt;/sup&gt;−1, and for the algebraic factors of 13&lt;sup&gt;564&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&amp;Exp=564&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 13&lt;sup&gt;564&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=564&amp;c0=-&amp;EN=&amp;LM=
* the 3180th Athena prime in base 13, 1B&lt;sub&gt;576&lt;/sub&gt;, ''N''−1 is 23×''R''&lt;sub&gt;576&lt;/sub&gt;(13), thus factor ''N''−1 is equivalent to factor the Cunningham number 13&lt;sup&gt;576&lt;/sup&gt;−1, and for the algebraic factors of 13&lt;sup&gt;576&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&amp;Exp=576&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 13&lt;sup&gt;576&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=576&amp;c0=-&amp;EN=&amp;LM=
* the 10320th Athena prime in base 17, 9&lt;sub&gt;292&lt;/sub&gt;1, ''N''−1 is 153×''R''&lt;sub&gt;292&lt;/sub&gt;(17), thus factor ''N''−1 is equivalent to factor the Cunningham number 17&lt;sup&gt;292&lt;/sup&gt;−1, and for the algebraic factors of 17&lt;sup&gt;292&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=17&amp;Exp=292&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 17&lt;sup&gt;292&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=17&amp;Exp=292&amp;c0=-&amp;EN=&amp;LM=
* the 13304th Athena prime in base 21, 7&lt;sub&gt;230&lt;/sub&gt;1, ''N''−1 is 147×''R''&lt;sub&gt;230&lt;/sub&gt;(21), thus factor ''N''−1 is equivalent to factor the Cunningham number 21&lt;sup&gt;230&lt;/sup&gt;−1, and for the algebraic factors of 21&lt;sup&gt;230&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=21&amp;Exp=230&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 21&lt;sup&gt;230&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=21&amp;Exp=230&amp;c0=-&amp;EN=&amp;LM=
* the 13355th Athena prime in base 21, 3&lt;sub&gt;1063&lt;/sub&gt;2, ''N''+1 is 3×''R''&lt;sub&gt;1064&lt;/sub&gt;(21), thus factor ''N''−1 is equivalent to factor the Cunningham number 21&lt;sup&gt;1064&lt;/sup&gt;−1, and for the algebraic factors of 21&lt;sup&gt;1064&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=21&amp;Exp=1064&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 21&lt;sup&gt;1064&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=21&amp;Exp=1064&amp;c0=-&amp;EN=&amp;LM=
* the 25199th Athena prime in base 26, 9K&lt;sub&gt;343&lt;/sub&gt;AP, ''N''+1 is 6370×''R''&lt;sub&gt;344&lt;/sub&gt;(26), thus factor ''N''+1 is equivalent to factor the Cunningham number 26&lt;sup&gt;344&lt;/sup&gt;−1, and for the algebraic factors of 26&lt;sup&gt;344&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&amp;Exp=344&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 26&lt;sup&gt;344&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&amp;Exp=344&amp;c0=-&amp;EN=&amp;LM=
* the 25200th Athena prime in base 26, 8&lt;sub&gt;354&lt;/sub&gt;1, ''N''−1 is 208×''R''&lt;sub&gt;354&lt;/sub&gt;(26), thus factor ''N''−1 is equivalent to factor the Cunningham number 26&lt;sup&gt;354&lt;/sup&gt;−1, and for the algebraic factors of 26&lt;sup&gt;354&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&amp;Exp=354&amp;LBIDPMList=A&amp;LBIDLODList=D, and for the prime factorization of 26&lt;sup&gt;354&lt;/sup&gt;−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&amp;Exp=354&amp;c0=-&amp;EN=&amp;LM=

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 Athena prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 Athena primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 Athena primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 Athena primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}

Base 6: 11 Athena primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 Athena primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331}

Base 8: 75 Athena primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}

Base 9: 151 Athena primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011}

Base 10: 77 Athena primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

Base 11: 1068 Athena (probable) primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena (probable) primes base 11]

Base 12: 106 Athena primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 Athena (probable) primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena (probable) primes base 13]

Base 14: 650 Athena primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena primes base 14]

Base 15: 1284 Athena primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena primes base 15]

Base 16: 2347 Athena (probable) primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena (probable) primes base 16]

Base 17: 10415 known Athena (probable) primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of known Athena (probable) primes base 17]

Base 18: 549 Athena primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena primes base 18]

Base 19: 31417 known Athena (probable) primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of known Athena (probable) primes base 19]

Base 20: 3314 Athena primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena primes base 20]

Base 21: 13386 known Athena (probable) primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of known Athena (probable) primes base 21]

Base 22: 8003 Athena (probable) primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena (probable) primes base 22]

Base 23: 65178 known Athena (probable) primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of known Athena (probable) primes base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for Athena problem base 23]

Base 24: 3409 Athena primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena primes base 24]

Base 25: 133639 known Athena (probable) primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of known Athena (probable) primes base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for Athena problem base 25]

Base 26: 25256 known Athena (probable) primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of known Athena (probable) primes base 26]

Base 27: 102852 known Athena (probable) primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of known Athena (probable) primes base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for Athena problem base 27]

Base 28: 25528 known Athena (probable) primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of known Athena (probable) primes base 28]

Base 29: 355242 known Athena (probable) primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of known Athena (probable) primes base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for Athena problem base 29]

Base 30: 2619 Athena (probable) primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena (probable) primes base 30]

Base 31: 569323 known Athena (probable) primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of known Athena (probable) primes base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for Athena problem base 31]

Base 32: 168882 known Athena (probable) primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of known Athena (probable) primes base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for Athena problem base 32]

Base 33: 280012 known Athena (probable) primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of known Athena (probable) primes base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for Athena problem base 33]

Base 34: 184785 known Athena (probable) primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of known Athena (probable) primes base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for Athena problem base 34]

Base 35: 720002 known Athena (probable) primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of known Athena (probable) primes base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for Athena problem base 35]

Base 36: 35286 known Athena (probable) primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of known Athena (probable) primes base 36]

== Condensed table for bases 2 ≤ ''b'' ≤ 36 ==

{|class=&quot;wikitable&quot;
||''b''||number of Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||base-''b'' form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' (write &quot;''d''&lt;sub&gt;''n''&lt;/sub&gt;&quot; if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||length of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' in decimal||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||''factordb'' entry of the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b''||the top 10 known Athena primes (or probable primes, which are Athena primes assuming their primality) in base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of unsolved families in the Athena problem in base ''b'' (all of these left families are linear families)||searching limit of length for the unsolved families in the Athena problem in base ''b'' (if there are different searching limits for the unsolved families in the Athena problem in base ''b'', choose the lowest searching limit)||
|-
||2||1||11||2||1||3||http://factordb.com/index.php?id=3&amp;open=ecm||http://factordb.com/index.php?showid=3&amp;base=2||0||–||
|-
||3||3||111&lt;br&gt;21&lt;br&gt;12||3&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;1&lt;br&gt;1||13&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=13&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=3&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=3&lt;nowiki/&gt;||0||–||
|-
||4||5||221&lt;br&gt;31&lt;br&gt;23&lt;br&gt;13&lt;br&gt;11||3&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;1&lt;br&gt;1||41&lt;br&gt;13&lt;br&gt;11&lt;br&gt;7&lt;br&gt;5||http://factordb.com/index.php?id=41&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=7&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=5&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=41&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=7&amp;base=4&lt;br&gt;http://factordb.com/index.php?showid=5&amp;base=4&lt;nowiki/&gt;||0||–||
|-
||5||22||10&lt;sub&gt;93&lt;/sub&gt;13&lt;br&gt;300031&lt;br&gt;44441&lt;br&gt;33331&lt;br&gt;33001&lt;br&gt;30301&lt;br&gt;14444&lt;br&gt;10103&lt;br&gt;3101&lt;br&gt;414||96&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4&lt;br&gt;3||67&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;3&lt;br&gt;3&lt;br&gt;3||5&lt;sup&gt;95&lt;/sup&gt;+8&lt;br&gt;9391&lt;br&gt;3121&lt;br&gt;2341&lt;br&gt;2251&lt;br&gt;1951&lt;br&gt;1249&lt;br&gt;653&lt;br&gt;401&lt;br&gt;109||http://factordb.com/index.php?id=1100000000034686071&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9391&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=3121&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2341&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2251&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1951&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1249&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=653&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=401&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=109&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000034686071&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=9391&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=3121&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2341&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=2251&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1951&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=1249&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=653&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=401&amp;base=5&lt;br&gt;http://factordb.com/index.php?showid=109&amp;base=5&lt;nowiki/&gt;||0||–||
|-
||6||11||40041&lt;br&gt;4441&lt;br&gt;4401&lt;br&gt;51&lt;br&gt;45&lt;br&gt;35&lt;br&gt;31&lt;br&gt;25&lt;br&gt;21&lt;br&gt;15||5&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||4&lt;br&gt;4&lt;br&gt;4&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2&lt;br&gt;2||5209&lt;br&gt;1033&lt;br&gt;1009&lt;br&gt;31&lt;br&gt;29&lt;br&gt;23&lt;br&gt;19&lt;br&gt;17&lt;br&gt;13&lt;br&gt;11||http://factordb.com/index.php?id=5209&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1033&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1009&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=31&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=29&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=23&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=19&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=17&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=13&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=5209&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1033&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=1009&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=31&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=29&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=23&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=19&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=17&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=13&amp;base=6&lt;br&gt;http://factordb.com/index.php?showid=11&amp;base=6&lt;nowiki/&gt;||0||–||
|-
||7||71||3&lt;sub&gt;16&lt;/sub&gt;1&lt;br&gt;510&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;3&lt;sub&gt;6&lt;/sub&gt;01&lt;br&gt;1100021&lt;br&gt;531101&lt;br&gt;351101&lt;br&gt;300053&lt;br&gt;150001&lt;br&gt;100121&lt;br&gt;40054||17&lt;br&gt;10&lt;br&gt;8&lt;br&gt;7&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;6&lt;br&gt;5||15&lt;br&gt;9&lt;br&gt;7&lt;br&gt;6&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;5&lt;br&gt;4||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2&lt;br&gt;36×7&lt;sup&gt;8&lt;/sup&gt;+1&lt;br&gt;(7&lt;sup&gt;8&lt;/sup&gt;−47)/2&lt;br&gt;134471&lt;br&gt;91631&lt;br&gt;62819&lt;br&gt;50459&lt;br&gt;28813&lt;br&gt;16871&lt;br&gt;9643||http://factordb.com/index.php?id=116315256993601&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=207532837&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=2882377&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=134471&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=91631&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=62819&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=50459&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=28813&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=16871&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=9643&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=116315256993601&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=207532837&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=2882377&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=134471&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=91631&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=62819&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=50459&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=28813&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=16871&amp;base=7&lt;br&gt;http://factordb.com/index.php?showid=9643&amp;base=7&lt;nowiki/&gt;||0||–||
|-
||8||75||4&lt;sub&gt;220&lt;/sub&gt;7&lt;br&gt;5&lt;sub&gt;13&lt;/sub&gt;25&lt;br&gt;7&lt;sub&gt;12&lt;/sub&gt;1&lt;br&gt;77774&lt;sub&gt;6&lt;/sub&gt;1&lt;br&gt;74&lt;sub&gt;7&lt;/sub&gt;1&lt;br&gt;4&lt;sub&gt;8&lt;/sub&gt;1&lt;br&gt;5&lt;sub&gt;5&lt;/sub&gt;025&lt;br&gt;5550525&lt;br&gt;5500525&lt;br&gt;4&lt;sub&gt;5&lt;/sub&gt;77||221&lt;br&gt;15&lt;br&gt;13&lt;br&gt;11&lt;br&gt;9&lt;br&gt;9&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||200&lt;br&gt;14&lt;br&gt;12&lt;br&gt;10&lt;br&gt;9&lt;br&gt;8&lt;br&gt;8&lt;br&gt;7&lt;br&gt;7&lt;br&gt;7||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7&lt;br&gt;(5×8&lt;sup&gt;15&lt;/sup&gt;−173)/7&lt;br&gt;8&lt;sup&gt;13&lt;/sup&gt;−7&lt;br&gt;(28669×8&lt;sup&gt;7&lt;/sup&gt;−25)/7&lt;br&gt;(53×8&lt;sup&gt;8&lt;/sup&gt;−25)/7&lt;br&gt;(4×8&lt;sup&gt;9&lt;/sup&gt;−25)/7&lt;br&gt;(5×8&lt;sup&gt;8&lt;/sup&gt;−2413)/7&lt;br&gt;1495381&lt;br&gt;1474901&lt;br&gt;(4×8&lt;sup&gt;7&lt;/sup&gt;+185)/7||http://factordb.com/index.php?id=1100000000416605822&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=25131694349141&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=549755813881&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=8589035809&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=127027489&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=76695841&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=11983381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1495381&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1474901&amp;open=ecm&lt;br&gt;http://factordb.com/index.php?id=1198399&amp;open=ecm&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000000416605822&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=25131694349141&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=549755813881&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=8589035809&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=127027489&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=76695841&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=11983381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1495381&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1474901&amp;base=8&lt;br&gt;http://factordb.com/index.php?showid=1198399&amp;base=8&lt;nowiki/&gt;||0||–||
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||33||280012~280093||DP&lt;sub&gt;95093&lt;/sub&gt;M5&lt;br&gt;HJ0&lt;sub&gt;94295&lt;/sub&gt;J&lt;br&gt;90&lt;sub&gt;93597&lt;/sub&gt;Q&lt;br&gt;9F0&lt;sub&gt;93157&lt;/sub&gt;N&lt;br&gt;7&lt;sub&gt;89449&lt;/sub&gt;333H&lt;br&gt;K3&lt;sub&gt;80751&lt;/sub&gt;6K&lt;br&gt;D&lt;sub&gt;80107&lt;/sub&gt;9UD&lt;br&gt;VFU&lt;sub&gt;72204&lt;/sub&gt;FK&lt;br&gt;J&lt;sub&gt;68715&lt;/sub&gt;2BJ&lt;br&gt;DF0&lt;sub&gt;68367&lt;/sub&gt;J||95096&lt;br&gt;94298&lt;br&gt;93599&lt;br&gt;93160&lt;br&gt;89453&lt;br&gt;80754&lt;br&gt;80110&lt;br&gt;72208&lt;br&gt;68718&lt;br&gt;68370||144405&lt;br&gt;143193&lt;br&gt;142131&lt;br&gt;141465&lt;br&gt;135835&lt;br&gt;122626&lt;br&gt;121648&lt;br&gt;109649&lt;br&gt;104350&lt;br&gt;103821||(441×33&lt;sup&gt;95095&lt;/sup&gt;−3833)/32&lt;br&gt;580×33&lt;sup&gt;94296&lt;/sup&gt;+19&lt;br&gt;9×33&lt;sup&gt;93598&lt;/sup&gt;+26&lt;br&gt;312×33&lt;sup&gt;93158&lt;/sup&gt;+23&lt;br&gt;(7×33&lt;sup&gt;89453&lt;/sup&gt;−4743239)/32&lt;br&gt;(643×33&lt;sup&gt;80753&lt;/sup&gt;+3709)/32&lt;br&gt;(13×33&lt;sup&gt;80110&lt;/sup&gt;−121453)/32&lt;br&gt;(16623×33&lt;sup&gt;72206&lt;/sup&gt;−8095)/16&lt;br&gt;(19×33&lt;sup&gt;68718&lt;/sup&gt;−600883)/32&lt;br&gt;444×33&lt;sup&gt;68368&lt;/sup&gt;+19||http://factordb.com/index.php?id=1100000005652348775&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652362811&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652375073&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652389776&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652430746&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652446200&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000005652461592&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614764298&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614770536&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004614784274&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000005652348775&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652362811&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652375073&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652389776&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652430746&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652446200&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000005652461592&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614764298&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614770536&amp;base=33&lt;br&gt;http://factordb.com/index.php?showid=1100000004614784274&amp;base=33&lt;nowiki/&gt;||81||100000||
|-
||34||184785~184832||GFGC&lt;sub&gt;99996&lt;/sub&gt;5&lt;br&gt;90&lt;sub&gt;97950&lt;/sub&gt;FJ&lt;br&gt;NM0&lt;sub&gt;85218&lt;/sub&gt;KX&lt;br&gt;F&lt;sub&gt;83189&lt;/sub&gt;H2HP&lt;br&gt;P&lt;sub&gt;79441&lt;/sub&gt;444P&lt;br&gt;6&lt;sub&gt;77027&lt;/sub&gt;8X&lt;br&gt;XQIQ&lt;sub&gt;72241&lt;/sub&gt;D&lt;br&gt;T&lt;sub&gt;66530&lt;/sub&gt;IF&lt;br&gt;4&lt;sub&gt;66152&lt;/sub&gt;B&lt;br&gt;2EEC&lt;sub&gt;66039&lt;/sub&gt;7||100000&lt;br&gt;97953&lt;br&gt;85222&lt;br&gt;83193&lt;br&gt;79445&lt;br&gt;77029&lt;br&gt;72245&lt;br&gt;66532&lt;br&gt;66153&lt;br&gt;66043||153148&lt;br&gt;150013&lt;br&gt;130516&lt;br&gt;127408&lt;br&gt;121669&lt;br&gt;117968&lt;br&gt;110642&lt;br&gt;101893&lt;br&gt;101312&lt;br&gt;101143||(209246×34&lt;sup&gt;99997&lt;/sup&gt;−81)/11&lt;br&gt;9×34&lt;sup&gt;97952&lt;/sup&gt;+529&lt;br&gt;804×34&lt;sup&gt;85220&lt;/sup&gt;+713&lt;br&gt;(5×34&lt;sup&gt;83193&lt;/sup&gt;+700233)/11&lt;br&gt;(25×34&lt;sup&gt;79445&lt;/sup&gt;−28062367)/33&lt;br&gt;(2×34&lt;sup&gt;77029&lt;/sup&gt;+1043)/11&lt;br&gt;(1288676×34&lt;sup&gt;72242&lt;/sup&gt;−455)/33&lt;br&gt;(29×34&lt;sup&gt;66532&lt;/sup&gt;−12833)/33&lt;br&gt;(4×34&lt;sup&gt;66153&lt;/sup&gt;+227)/33&lt;br&gt;(30826×34&lt;sup&gt;66040&lt;/sup&gt;−59)/11||http://factordb.com/index.php?id=1100000004702891268&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702894713&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702900996&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702910376&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702913746&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004702918600&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399656529&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399657696&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399658651&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004399659716&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000004702891268&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702894713&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702900996&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702910376&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702913746&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004702918600&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399656529&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399657696&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399658651&amp;base=34&lt;br&gt;http://factordb.com/index.php?showid=1100000004399659716&amp;base=34&lt;nowiki/&gt;||47||100000||
|-
||35||720002~720062||N0N&lt;sub&gt;99971&lt;/sub&gt;9&lt;br&gt;V0&lt;sub&gt;83669&lt;/sub&gt;E73&lt;br&gt;N&lt;sub&gt;81563&lt;/sub&gt;K7N&lt;br&gt;BJ0&lt;sub&gt;81279&lt;/sub&gt;N&lt;br&gt;J0&lt;sub&gt;80062&lt;/sub&gt;FUH&lt;br&gt;43V&lt;sub&gt;79754&lt;/sub&gt;&lt;br&gt;9&lt;sub&gt;76600&lt;/sub&gt;K3&lt;br&gt;LB&lt;sub&gt;71366&lt;/sub&gt;PB&lt;br&gt;Q&lt;sub&gt;64150&lt;/sub&gt;H&lt;br&gt;50&lt;sub&gt;63397&lt;/sub&gt;5R||99974&lt;br&gt;83673&lt;br&gt;81566&lt;br&gt;81282&lt;br&gt;80066&lt;br&gt;79756&lt;br&gt;76602&lt;br&gt;71369&lt;br&gt;64151&lt;br&gt;63400||154367&lt;br&gt;129197&lt;br&gt;125944&lt;br&gt;125505&lt;br&gt;123628&lt;br&gt;123148&lt;br&gt;118279&lt;br&gt;110199&lt;br&gt;99054&lt;br&gt;97894||(27393×35&lt;sup&gt;99972&lt;/sup&gt;−499)/34&lt;br&gt;31×35&lt;sup&gt;83672&lt;/sup&gt;+17398&lt;br&gt;(23×35&lt;sup&gt;81566&lt;/sup&gt;−144013)/34&lt;br&gt;404×35&lt;sup&gt;81280&lt;/sup&gt;+23&lt;br&gt;19×35&lt;sup&gt;80065&lt;/sup&gt;+19442&lt;br&gt;(4893×35&lt;sup&gt;79754&lt;/sup&gt;−31)/34&lt;br&gt;(9×35&lt;sup&gt;76602&lt;/sup&gt;+12877)/34&lt;br&gt;(725×35&lt;sup&gt;71368&lt;/sup&gt;+16649)/34&lt;br&gt;(13×35&lt;sup&gt;64151&lt;/sup&gt;−166)/17&lt;br&gt;5×35&lt;sup&gt;63399&lt;/sup&gt;+202||http://factordb.com/index.php?id=1100000008248342445&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248353306&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248375642&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248397018&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248412468&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248418540&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008248423670&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192119974&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192126630&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000008192129294&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000008248342445&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248353306&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248375642&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248397018&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248412468&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248418540&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008248423670&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192119974&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192126630&amp;base=35&lt;br&gt;http://factordb.com/index.php?showid=1100000008192129294&amp;base=35&lt;nowiki/&gt;||60||100000||
|-
||36||35286~35290||P&lt;sub&gt;81993&lt;/sub&gt;SZ&lt;br&gt;S0&lt;sub&gt;75007&lt;/sub&gt;8H&lt;br&gt;7K&lt;sub&gt;26567&lt;/sub&gt;Z&lt;br&gt;J&lt;sub&gt;10117&lt;/sub&gt;LJ&lt;br&gt;VL0&lt;sub&gt;7258&lt;/sub&gt;J&lt;br&gt;EO0&lt;sub&gt;6177&lt;/sub&gt;V&lt;br&gt;FZ&lt;sub&gt;5777&lt;/sub&gt;3P&lt;br&gt;T09&lt;sub&gt;4618&lt;/sub&gt;1&lt;br&gt;RY&lt;sub&gt;4562&lt;/sub&gt;H&lt;br&gt;OZ&lt;sub&gt;3932&lt;/sub&gt;AZ||81995&lt;br&gt;75010&lt;br&gt;26569&lt;br&gt;10119&lt;br&gt;7261&lt;br&gt;6180&lt;br&gt;5780&lt;br&gt;4621&lt;br&gt;4564&lt;br&gt;3935||127609&lt;br&gt;116739&lt;br&gt;41349&lt;br&gt;15748&lt;br&gt;11301&lt;br&gt;9618&lt;br&gt;8996&lt;br&gt;7192&lt;br&gt;7103&lt;br&gt;6124||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7&lt;br&gt;28×36&lt;sup&gt;75009&lt;/sup&gt;+305&lt;br&gt;(53×36&lt;sup&gt;26568&lt;/sup&gt;+101)/7&lt;br&gt;(19×36&lt;sup&gt;10119&lt;/sup&gt;+2501)/35&lt;br&gt;1137×36&lt;sup&gt;7259&lt;/sup&gt;+19&lt;br&gt;528×36&lt;sup&gt;6178&lt;/sup&gt;+31&lt;br&gt;16×36&lt;sup&gt;5779&lt;/sup&gt;−1163&lt;br&gt;(36549×36&lt;sup&gt;4619&lt;/sup&gt;−289)/35&lt;br&gt;(979×36&lt;sup&gt;4563&lt;/sup&gt;−629)/35&lt;br&gt;25×36&lt;sup&gt;3934&lt;/sup&gt;−901||http://factordb.com/index.php?id=1100000002394962083&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000004020085177&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003896952461&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362491&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362489&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362488&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362487&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362486&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000003807362485&amp;open=prime&lt;br&gt;http://factordb.com/index.php?id=1100000000840634476&amp;open=prime&lt;nowiki/&gt;||http://factordb.com/index.php?showid=1100000002394962083&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000004020085177&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003896952461&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362491&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362489&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362488&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362487&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362486&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000003807362485&amp;base=36&lt;br&gt;http://factordb.com/index.php?showid=1100000000840634476&amp;base=36&lt;nowiki/&gt;||4||200000||
|}

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the Athena primes, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
      <sha1>rsbxzt0hsy8lqj9bvifsh6wsjbe1go9</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Original research and scholarly standards</title>
    <ns>4</ns>
    <id>329789</id>
    <revision>
      <id>2829720</id>
      <parentid>2813305</parentid>
      <timestamp>2026-08-30T13:25:34Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* Process */ tweak</comment>
      <origin>2829720</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5330" sha1="60s3o7y7ogohy65o4ju91weuns5ic1n" xml:space="preserve">{{merge|Wikiversity:Original research}}
{{proposal}}
{{info|This is a rough draft}}
''Purpose'': Explanation of acceptable/unacceptable original research, bridging the gap of all Wikiversity pages that detail research, research standards, and commitment to academic learning and growth. 

Wikiversity, as a part of the Wikimedia Foundation, has a commitment to [[Wikiversity:Learning by doing|active]] [[Wikiversity:Learning|learning]] that is in adherence to proper, high-quality research standards. This includes adhering to [[Wikiversity:Research guidelines#Ethics|scholarly ethics]] when conducting and presenting research, which includes:

* subjecting research to [[Wikiversity:Peer review|peer review]] and independent verification, whichever and wherever its applicable.
* not violating established ethical guidelines
* honesty, including transparency, listing objectives, [[Wikiversity:Verifiability|citing reliable sources]], and making note of any [[Wikiversity:Disclosures|notable disclosures]]. See [[Wikiversity:Research_process#What_are_research_ethics?]] for more research ethics that editors should strive for. 

Wikiversity, as part of its [[Wikiversity:What is Wikiversity?|aims]], encourages &quot;learning by doing&quot;, or [[Wikiversity:Developing Wikiversity through action research|through active research]]. Unlike its counterpart, [[Wikiversity:Wikipedia|Wikipedia]], Wikiversity allows [[Wikiversity:Original research|original research.]] Briefly, original research examples include:

# Conducting an experiment testing the waterproofability of three brand wallets.
# [[Help:Lab reports|Lab reports]] conducted in a scientific venue, such as [[Help:Assignment|class assignments]]. See examples of lab reports in this category: [[:Category:Lab reports]]. 
# Academic essays as part of homework assignments. See examples of essays here: [[:Category:Essays]].

While Wikiversity enjoys the benefit of flexibility, Wikiversity strives to avoid content that promotes severe deviations from mainstream science, and speculative theories that present themselves as established science without proper contextualization/disclosure (such as the promotion of [[wikipedia:Fringe_theory|fringe theories]] or [[wikipedia:Pseudoscience|pseudoscience]]). Such content harms not only the reputation of Wikiversity but also the ability of viewers and collaborators to properly engage and foster a learning environment&lt;ref&gt;''Using Wikiversity as an academic discussion forum may help share ideas that may promote research and learning.'' - [[Wikiversity:What_is_Wikiversity?#Wikiversity_for_sharing_materials,_ideas,_community]]&lt;/ref&gt;. 

Original research '''must''' be presented in a way that readers can understand that the authors are presenting new ideas that are in accordance with scientific practices, including honest disclosures (including NPOV), distinction between speculation and established knowledge, and invitation of peer review (through the [[Template:To be peer reviewed|to be peer reviewed]] template; though this does not serve as a &quot;green light&quot; for fringe research)&lt;ref&gt;would require changing [[Wikiversity:Original research]] requirements.&lt;/ref&gt;. 

Research that deviates from standard scientific practices includes presenting scientific theories without reliable sources backing them up, using or manipulating scientific terminology, making extreme claims, or failing to provide a clear learning structure revolving around [[Wikiversity:Research collaboration|collaborative learning]]. Pages that fail to meet these requirements may be moved to userspace/draft, require cleanup notices, or be heavily rewritten as they do not meet Wikiversity's [[Wikiversity:Scope|scope]]&lt;ref&gt;''Wikiversity offers a collaborative environment for the creation, sharing, and discussion of [[open educational resources]], [[open research]] and [[open academia]].'' ([[Wikiversity:Scope]]) - derived from this.&lt;/ref&gt;. 

The essential rule is to be honest with your readers and to contextualize the learning resource you are presenting so viewers can extract as much learning value from your resource as possible. 

== Guidelines/Checklist ==
== Process ==
#All original research must first be submitted under the &quot;Draft&quot; namespace.
#Author of the page must submit it for mainspace consideration at ''[page name?]''
#A Wikiversity bureaucrat from the [[Wikiversity:Review board|review board]] must approve the page and confirm that the page is in accordance with Wikiversity's original research policy.
#If successful, the page may be moved into the mainspace.

== See also ==
'''Wikiversity Space Links'''

* [[Wikiversity:Scope]]

* [[Wikiversity:Original research]] (proposed policy)

* [[Wikiversity:Research]]
* [[Wikiversity:Research process]]
** [[Wikiversity:Research_process#What_are_research_ethics?]]
* [[Wikiversity:Scholarly ethics]]
* [[Wikiversity:Peer review]]
** [[Wikiversity:Peer review verification]]
* [[Wikiversity:Scope of research]]
* [[Wikiversity:POV]]
* [[Wikiversity:What is Wikiversity?]]
* [[Wikiversity:What Wikiversity is not]]

'''Pseudoscience/Fringe Theories'''
* [[wikipedia:Pseudoscience]]
* [[wikipedia:Wikipedia:Fringe_theories_for_dummies|Wikipedia:Fringe theories for dummies]]
* [[Wikipedia:Wikipedia:Fringe_theories#Pseudoscience]]

== Notes ==
&lt;references /&gt;
[[Category:Atcovi's Work]]</text>
      <sha1>60s3o7y7ogohy65o4ju91weuns5ic1n</sha1>
    </revision>
    <revision>
      <id>2829721</id>
      <parentid>2829720</parentid>
      <timestamp>2026-08-30T13:27:13Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* Process */ tweak</comment>
      <origin>2829721</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5312" sha1="reerpj3ue6629k8zryyvk3gbqdmyz0n" xml:space="preserve">{{merge|Wikiversity:Original research}}
{{proposal}}
{{info|This is a rough draft}}
''Purpose'': Explanation of acceptable/unacceptable original research, bridging the gap of all Wikiversity pages that detail research, research standards, and commitment to academic learning and growth. 

Wikiversity, as a part of the Wikimedia Foundation, has a commitment to [[Wikiversity:Learning by doing|active]] [[Wikiversity:Learning|learning]] that is in adherence to proper, high-quality research standards. This includes adhering to [[Wikiversity:Research guidelines#Ethics|scholarly ethics]] when conducting and presenting research, which includes:

* subjecting research to [[Wikiversity:Peer review|peer review]] and independent verification, whichever and wherever its applicable.
* not violating established ethical guidelines
* honesty, including transparency, listing objectives, [[Wikiversity:Verifiability|citing reliable sources]], and making note of any [[Wikiversity:Disclosures|notable disclosures]]. See [[Wikiversity:Research_process#What_are_research_ethics?]] for more research ethics that editors should strive for. 

Wikiversity, as part of its [[Wikiversity:What is Wikiversity?|aims]], encourages &quot;learning by doing&quot;, or [[Wikiversity:Developing Wikiversity through action research|through active research]]. Unlike its counterpart, [[Wikiversity:Wikipedia|Wikipedia]], Wikiversity allows [[Wikiversity:Original research|original research.]] Briefly, original research examples include:

# Conducting an experiment testing the waterproofability of three brand wallets.
# [[Help:Lab reports|Lab reports]] conducted in a scientific venue, such as [[Help:Assignment|class assignments]]. See examples of lab reports in this category: [[:Category:Lab reports]]. 
# Academic essays as part of homework assignments. See examples of essays here: [[:Category:Essays]].

While Wikiversity enjoys the benefit of flexibility, Wikiversity strives to avoid content that promotes severe deviations from mainstream science, and speculative theories that present themselves as established science without proper contextualization/disclosure (such as the promotion of [[wikipedia:Fringe_theory|fringe theories]] or [[wikipedia:Pseudoscience|pseudoscience]]). Such content harms not only the reputation of Wikiversity but also the ability of viewers and collaborators to properly engage and foster a learning environment&lt;ref&gt;''Using Wikiversity as an academic discussion forum may help share ideas that may promote research and learning.'' - [[Wikiversity:What_is_Wikiversity?#Wikiversity_for_sharing_materials,_ideas,_community]]&lt;/ref&gt;. 

Original research '''must''' be presented in a way that readers can understand that the authors are presenting new ideas that are in accordance with scientific practices, including honest disclosures (including NPOV), distinction between speculation and established knowledge, and invitation of peer review (through the [[Template:To be peer reviewed|to be peer reviewed]] template; though this does not serve as a &quot;green light&quot; for fringe research)&lt;ref&gt;would require changing [[Wikiversity:Original research]] requirements.&lt;/ref&gt;. 

Research that deviates from standard scientific practices includes presenting scientific theories without reliable sources backing them up, using or manipulating scientific terminology, making extreme claims, or failing to provide a clear learning structure revolving around [[Wikiversity:Research collaboration|collaborative learning]]. Pages that fail to meet these requirements may be moved to userspace/draft, require cleanup notices, or be heavily rewritten as they do not meet Wikiversity's [[Wikiversity:Scope|scope]]&lt;ref&gt;''Wikiversity offers a collaborative environment for the creation, sharing, and discussion of [[open educational resources]], [[open research]] and [[open academia]].'' ([[Wikiversity:Scope]]) - derived from this.&lt;/ref&gt;. 

The essential rule is to be honest with your readers and to contextualize the learning resource you are presenting so viewers can extract as much learning value from your resource as possible. 

== Guidelines/Checklist ==
== Process ==
#All original research must first be submitted under the &quot;Draft&quot; namespace.
#Author of the page must submit it for mainspace consideration at ''[page name?]''
#A member of the [[Wikiversity:Review board|review board]] must approve the page and confirm that the page is in accordance with Wikiversity's original research policy.
#If successful, the page may be moved into the mainspace.

== See also ==
'''Wikiversity Space Links'''

* [[Wikiversity:Scope]]

* [[Wikiversity:Original research]] (proposed policy)

* [[Wikiversity:Research]]
* [[Wikiversity:Research process]]
** [[Wikiversity:Research_process#What_are_research_ethics?]]
* [[Wikiversity:Scholarly ethics]]
* [[Wikiversity:Peer review]]
** [[Wikiversity:Peer review verification]]
* [[Wikiversity:Scope of research]]
* [[Wikiversity:POV]]
* [[Wikiversity:What is Wikiversity?]]
* [[Wikiversity:What Wikiversity is not]]

'''Pseudoscience/Fringe Theories'''
* [[wikipedia:Pseudoscience]]
* [[wikipedia:Wikipedia:Fringe_theories_for_dummies|Wikipedia:Fringe theories for dummies]]
* [[Wikipedia:Wikipedia:Fringe_theories#Pseudoscience]]

== Notes ==
&lt;references /&gt;
[[Category:Atcovi's Work]]</text>
      <sha1>reerpj3ue6629k8zryyvk3gbqdmyz0n</sha1>
    </revision>
    <revision>
      <id>2829722</id>
      <parentid>2829721</parentid>
      <timestamp>2026-08-30T13:27:44Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* Process */ small note for later</comment>
      <origin>2829722</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5392" sha1="3c15ryl9ol782xndqbs93zczvlllphw" xml:space="preserve">{{merge|Wikiversity:Original research}}
{{proposal}}
{{info|This is a rough draft}}
''Purpose'': Explanation of acceptable/unacceptable original research, bridging the gap of all Wikiversity pages that detail research, research standards, and commitment to academic learning and growth. 

Wikiversity, as a part of the Wikimedia Foundation, has a commitment to [[Wikiversity:Learning by doing|active]] [[Wikiversity:Learning|learning]] that is in adherence to proper, high-quality research standards. This includes adhering to [[Wikiversity:Research guidelines#Ethics|scholarly ethics]] when conducting and presenting research, which includes:

* subjecting research to [[Wikiversity:Peer review|peer review]] and independent verification, whichever and wherever its applicable.
* not violating established ethical guidelines
* honesty, including transparency, listing objectives, [[Wikiversity:Verifiability|citing reliable sources]], and making note of any [[Wikiversity:Disclosures|notable disclosures]]. See [[Wikiversity:Research_process#What_are_research_ethics?]] for more research ethics that editors should strive for. 

Wikiversity, as part of its [[Wikiversity:What is Wikiversity?|aims]], encourages &quot;learning by doing&quot;, or [[Wikiversity:Developing Wikiversity through action research|through active research]]. Unlike its counterpart, [[Wikiversity:Wikipedia|Wikipedia]], Wikiversity allows [[Wikiversity:Original research|original research.]] Briefly, original research examples include:

# Conducting an experiment testing the waterproofability of three brand wallets.
# [[Help:Lab reports|Lab reports]] conducted in a scientific venue, such as [[Help:Assignment|class assignments]]. See examples of lab reports in this category: [[:Category:Lab reports]]. 
# Academic essays as part of homework assignments. See examples of essays here: [[:Category:Essays]].

While Wikiversity enjoys the benefit of flexibility, Wikiversity strives to avoid content that promotes severe deviations from mainstream science, and speculative theories that present themselves as established science without proper contextualization/disclosure (such as the promotion of [[wikipedia:Fringe_theory|fringe theories]] or [[wikipedia:Pseudoscience|pseudoscience]]). Such content harms not only the reputation of Wikiversity but also the ability of viewers and collaborators to properly engage and foster a learning environment&lt;ref&gt;''Using Wikiversity as an academic discussion forum may help share ideas that may promote research and learning.'' - [[Wikiversity:What_is_Wikiversity?#Wikiversity_for_sharing_materials,_ideas,_community]]&lt;/ref&gt;. 

Original research '''must''' be presented in a way that readers can understand that the authors are presenting new ideas that are in accordance with scientific practices, including honest disclosures (including NPOV), distinction between speculation and established knowledge, and invitation of peer review (through the [[Template:To be peer reviewed|to be peer reviewed]] template; though this does not serve as a &quot;green light&quot; for fringe research)&lt;ref&gt;would require changing [[Wikiversity:Original research]] requirements.&lt;/ref&gt;. 

Research that deviates from standard scientific practices includes presenting scientific theories without reliable sources backing them up, using or manipulating scientific terminology, making extreme claims, or failing to provide a clear learning structure revolving around [[Wikiversity:Research collaboration|collaborative learning]]. Pages that fail to meet these requirements may be moved to userspace/draft, require cleanup notices, or be heavily rewritten as they do not meet Wikiversity's [[Wikiversity:Scope|scope]]&lt;ref&gt;''Wikiversity offers a collaborative environment for the creation, sharing, and discussion of [[open educational resources]], [[open research]] and [[open academia]].'' ([[Wikiversity:Scope]]) - derived from this.&lt;/ref&gt;. 

The essential rule is to be honest with your readers and to contextualize the learning resource you are presenting so viewers can extract as much learning value from your resource as possible. 

== Guidelines/Checklist ==
== Process ==
#All original research must first be submitted under the &quot;Draft&quot; namespace.
#Author of the page must submit it for mainspace consideration at ''[page name?]''
#A member of the [[Wikiversity:Review board|review board]]&lt;ref&gt;wouldn't this be a Wikiversity 'crat? Or best be a Wikiversity 'crat?&lt;/ref&gt; must approve the page and confirm that the page is in accordance with Wikiversity's original research policy.
#If successful, the page may be moved into the mainspace.

== See also ==
'''Wikiversity Space Links'''

* [[Wikiversity:Scope]]

* [[Wikiversity:Original research]] (proposed policy)

* [[Wikiversity:Research]]
* [[Wikiversity:Research process]]
** [[Wikiversity:Research_process#What_are_research_ethics?]]
* [[Wikiversity:Scholarly ethics]]
* [[Wikiversity:Peer review]]
** [[Wikiversity:Peer review verification]]
* [[Wikiversity:Scope of research]]
* [[Wikiversity:POV]]
* [[Wikiversity:What is Wikiversity?]]
* [[Wikiversity:What Wikiversity is not]]

'''Pseudoscience/Fringe Theories'''
* [[wikipedia:Pseudoscience]]
* [[wikipedia:Wikipedia:Fringe_theories_for_dummies|Wikipedia:Fringe theories for dummies]]
* [[Wikipedia:Wikipedia:Fringe_theories#Pseudoscience]]

== Notes ==
&lt;references /&gt;
[[Category:Atcovi's Work]]</text>
      <sha1>3c15ryl9ol782xndqbs93zczvlllphw</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Cancer screening and emotion</title>
    <ns>0</ns>
    <id>330080</id>
    <revision>
      <id>2829770</id>
      <parentid>2829554</parentid>
      <timestamp>2026-08-30T20:23:02Z</timestamp>
      <contributor>
        <username>U3280825</username>
        <id>3110195</id>
      </contributor>
      <origin>2829770</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6917" sha1="rdxjf52xzzqg2vz4ar0ulo91uhw8hgj" xml:space="preserve">{{title|Cancer screening and emotion:&lt;br&gt;How do emotions such as fear, anxiety, and relief influence cancer screening uptake?
}}

__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:A picture is worth a thousand words.jpg|right|thumb|200px|'''Figure 1'''. Use a captioned image to illustrate the scenariokY AO Lo]]

Scenario
Megan, 50, discovers a letter in her mailbox inviting her a routine mammogram. She has continuously delayed it for the last four years. Her mother passed as a result of breast cancer, and the mere thought of the appointment makes her stomach knot. A part of her is comfortable with the uncertainty. The letter lays within a draw. In contrast, her friend Stella obtains the same letter a week later. Stella promptly schedules her appointment, Stella's view is synonymous with taking responsibility of her health. The procedure is the same, the same risk is attributed. Consequently, they feel completely differently about it and make opposite decisions.
{{RoundBoxBottom}}

Cancer screening programs for instance mammography, cervical screening, and bowl screening can significantly reduce mortality through early detection. Conversely, engagement remains below established national targets in Australia and internationally (Australian Institute of Health and Welfare, 2026). 

Key points this chapter will develop:
* Recognising these emotional dynamics attributes to screening programs to develop communication strategies that improve uptake.

* Individuals decision to participate in screening are not soley driven by rational evaluation; anxiety, fear, embarrassment, are all determinants as to whether individuals par take (Consedine et al., 2004)


{{RoundBoxTop|theme=2}}
'''Focus questions'''
* How can emotional determinants be addressed to improve screening participation? 
* What is the role of relief as an emotional driver in the uptake of screening?
* What emotions are experienced in relation to cancer screening?
* How does anxiety and fear act as barriers to screening uptake?
{{RoundBoxBottom}}

== What emotions are involved in cancer screening?==
* Emotional responses to screening comprise fear, anxiety, worry, stress, broadly conceptualised as anticipatory anxiety (Goodwin et al.,2023)
* Concern around the screening procedure and the cancer itself was reported between 10-50% of participants (Goodwin et al.,2023)
* A review of 74 studies identified that diagnostic anxiety tended to promote screening, conversely, procedural fear was almost reliably discouraging it (Goodwin et al., 2023).

==How does anxiety and fear motivate cancer screening?==
* Evidence from a systematic review illustrated that anxiety of a possible cancer diagnosis was often linked to greater screening uptake. Conversely, the fear of procedure discouraged participation (Goodwin et al.,2023).
* In a large Australian sample, screening engagement increased as an anxiety also increased but when anxiety was severe individuals were less likely to engage in screening (Anderson et al.,2023).
* Protection Motivation Theory proposes that individuals are highly probable to screen when they feel at harm and believe screening will benefit (Rogers, 1975).

==How does anxiety and fear act as barriers in cancer screening uptake?==
* Fear of receiving screening results can undermine participation, with fear of receiving a positive result as one of the biggest discouragers  of cervical screening among Australian women (Nagendiram et al., 2020).
* Severe anxiety may mitigate screening participation, with Australian literature illustrating lower bowel cancer screening participation across individuals experiencing severe anxiety (Anderson et al., 2023).
* Anticipatory anxiety associated around the screening procedure can attribute to avoidance, with never-screened or under screened Australian's noting psychological distress accompanied by invasive and sometimes discomfort-inducing nature of cervical screening (Creagh et al., 2021).

== How does relief influence cancer screening uptake?==
*

==Conclusion==
* The behavioural impacts of emotional responses on screening uptake, anxiety and fear proficient discouraging and motivating screening participation shaped by their focus and intensity
* Developing greater insight into the emotional components shaping screening decisions, guiding he development of strategies to mitigate psychological barriers promoting greater engagement
* Anxiety and fear influence screening in multifaceted ways, illustrating that emotional processes collectively shape screening behaviour.


==See also==
Anderson, L. E., Ireland, M. J., Myers, L., Avenell, C., Connaughton, T., &amp; Goodwin, B. C. (2023). Psychological distress and bowel cancer screening participation. Psycho-Oncology, 32(2), 229–236. https://doi.org/10.1002/pon.6072

Australian Institute of Health and Welfare. (2026). Cancer screening. Australian Government. https://www.aihw.gov.au/reports/australias-health/cancer-screening-and-treatment

Consedine, N. S., Magai, C., Krivoshekova, Y. S., Ryzewicz, L., &amp; Neugut, A. I. (2004). Fear, anxiety, worry, and breast cancer screening behavior: A critical review. Cancer Epidemiology, Biomarkers &amp; Prevention, 13(4), 501–510. https://doi.org/10.1158/1055-9965.501.13.4

Creagh, N. S., Zammit, C., Brotherton, J. M. L., Saville, M., McDermott, T., Nightingale, C., &amp; Kelaher, M. (2021). Self-collection cervical screening in the renewed National Cervical Screening Program: A qualitative study. Medical Journal of Australia, 215(8), 354–358. https://doi.org/10.5694/mja2.51137

Goodwin, B. C., Anderson, L., Collins, K., Sanjida, S., Riba, M., Singh, G. K., Campbell, K. M., Green, H., Ishaque, S., Kwok, A., Opozda, M. J., Pearn, A., Shaw, J., Sansom-Daly, U. M., Tsirgiotis, J. M., Janda, M., &amp; Grech, L. (2023). Anticipatory anxiety and participation in cancer screening: A systematic review. Psycho-Oncology, 32(12), 1773–1786. https://doi.org/10.1002/pon.6238

Lemmo, D., Martino, M. L., Vallone, F., Donizzetti, A. R., Freda, M. F., &amp; Caso, D. (2023). Clinical and psychosocial constructs for breast, cervical, and colorectal cancer screening participation: A systematic review. International Journal of Clinical and Health Psychology, 23(2), 100354. https://doi.org/10.1016/j.ijchp.2022.100354

Nagendiram, A., Bougher, H., Banks, J., Hall, L., &amp; Heal, C. (2020). Australian women's self-perceived barriers to participation in cervical cancer screening: A systematic review. Health Promotion Journal of Australia, 31(3), 343–353. https://doi.org/10.1002/hpja.280

Rogers, R. W. (1975). A protection motivation theory of fear appeals and attitude change. The Journal of Psychology, 91(1), 93–114. https://doi.org/10.1080/00223980.1975.9915803

Witte, K. (1992). Putting the fear back into fear appeals: The extended parallel process model. Communication Monographs, 59(4), 329–349. https://doi.org/10.1080/03637759209376276</text>
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}}

__TOC__

==Overview==
{{RoundBoxTop|theme=3}}

[[File:Woman receives mammogram (3).jpg|Woman_receives_mammogram_(3)]]

Figure 1. A woman undergoing a standard mammogram procedure. Image by Rhoda Baer, National Cancer Institute, public domain.]]


Scenario

Megan, 50, discovers a letter in her mailbox inviting her a routine mammogram. She has continuously delayed it for the last four years. Her mother passed as a result of breast cancer, and the mere thought of the appointment makes her stomach knot. A part of her is comfortable with the uncertainty. The letter lays within a draw. In contrast, her friend Stella obtains the same letter a week later. Stella promptly schedules her appointment, Stella's view is synonymous with taking responsibility of her health. The procedure is the same, the same risk is attributed. Consequently, they feel completely differently about it and make opposite decisions.
{{RoundBoxBottom}}

Cancer screening programs for instance mammography, cervical screening, and bowl screening can significantly reduce mortality through early detection. Conversely, engagement remains below established national targets in Australia and internationally (Australian Institute of Health and Welfare, 2026). 

Key points this chapter will develop:
* Recognising these emotional dynamics attributes to screening programs to develop communication strategies that improve uptake.

* Individuals decision to participate in screening are not soley driven by rational evaluation; anxiety, fear, embarrassment, are all determinants as to whether individuals par take (Consedine et al., 2004)


{{RoundBoxTop|theme=2}}
'''Focus questions'''
* How can emotional determinants be addressed to improve screening participation? 
* What is the role of relief as an emotional driver in the uptake of screening?
* What emotions are experienced in relation to cancer screening?
* How does anxiety and fear act as barriers to screening uptake?
{{RoundBoxBottom}}

== What emotions are involved in cancer screening?==
* Emotional responses to screening comprise fear, anxiety, worry, stress, broadly conceptualised as anticipatory anxiety (Goodwin et al.,2023)
* Concern around the screening procedure and the cancer itself was reported between 10-50% of participants (Goodwin et al.,2023)
* A review of 74 studies identified that diagnostic anxiety tended to promote screening, conversely, procedural fear was almost reliably discouraging it (Goodwin et al., 2023).

==How does anxiety and fear motivate cancer screening?==
* Evidence from a systematic review illustrated that anxiety of a possible cancer diagnosis was often linked to greater screening uptake. Conversely, the fear of procedure discouraged participation (Goodwin et al.,2023).
* In a large Australian sample, screening engagement increased as an anxiety also increased but when anxiety was severe individuals were less likely to engage in screening (Anderson et al.,2023).
* Protection Motivation Theory proposes that individuals are highly probable to screen when they feel at harm and believe screening will benefit (Rogers, 1975).

==How does anxiety and fear act as barriers in cancer screening uptake?==
* Fear of receiving screening results can undermine participation, with fear of receiving a positive result as one of the biggest discouragers  of cervical screening among Australian women (Nagendiram et al., 2020).
* Severe anxiety may mitigate screening participation, with Australian literature illustrating lower bowel cancer screening participation across individuals experiencing severe anxiety (Anderson et al., 2023).
* Anticipatory anxiety associated around the screening procedure can attribute to avoidance, with never-screened or under screened Australian's noting psychological distress accompanied by invasive and sometimes discomfort-inducing nature of cervical screening (Creagh et al., 2021).

== How does relief influence cancer screening uptake?==
*

==Conclusion==
* The behavioural impacts of emotional responses on screening uptake, anxiety and fear proficient discouraging and motivating screening participation shaped by their focus and intensity
* Developing greater insight into the emotional components shaping screening decisions, guiding he development of strategies to mitigate psychological barriers promoting greater engagement
* Anxiety and fear influence screening in multifaceted ways, illustrating that emotional processes collectively shape screening behaviour.


==See also==
Anderson, L. E., Ireland, M. J., Myers, L., Avenell, C., Connaughton, T., &amp; Goodwin, B. C. (2023). Psychological distress and bowel cancer screening participation. Psycho-Oncology, 32(2), 229–236. https://doi.org/10.1002/pon.6072

Australian Institute of Health and Welfare. (2026). Cancer screening. Australian Government. https://www.aihw.gov.au/reports/australias-health/cancer-screening-and-treatment

Consedine, N. S., Magai, C., Krivoshekova, Y. S., Ryzewicz, L., &amp; Neugut, A. I. (2004). Fear, anxiety, worry, and breast cancer screening behavior: A critical review. Cancer Epidemiology, Biomarkers &amp; Prevention, 13(4), 501–510. https://doi.org/10.1158/1055-9965.501.13.4

Creagh, N. S., Zammit, C., Brotherton, J. M. L., Saville, M., McDermott, T., Nightingale, C., &amp; Kelaher, M. (2021). Self-collection cervical screening in the renewed National Cervical Screening Program: A qualitative study. Medical Journal of Australia, 215(8), 354–358. https://doi.org/10.5694/mja2.51137

Goodwin, B. C., Anderson, L., Collins, K., Sanjida, S., Riba, M., Singh, G. K., Campbell, K. M., Green, H., Ishaque, S., Kwok, A., Opozda, M. J., Pearn, A., Shaw, J., Sansom-Daly, U. M., Tsirgiotis, J. M., Janda, M., &amp; Grech, L. (2023). Anticipatory anxiety and participation in cancer screening: A systematic review. Psycho-Oncology, 32(12), 1773–1786. https://doi.org/10.1002/pon.6238

Lemmo, D., Martino, M. L., Vallone, F., Donizzetti, A. R., Freda, M. F., &amp; Caso, D. (2023). Clinical and psychosocial constructs for breast, cervical, and colorectal cancer screening participation: A systematic review. International Journal of Clinical and Health Psychology, 23(2), 100354. https://doi.org/10.1016/j.ijchp.2022.100354

Nagendiram, A., Bougher, H., Banks, J., Hall, L., &amp; Heal, C. (2020). Australian women's self-perceived barriers to participation in cervical cancer screening: A systematic review. Health Promotion Journal of Australia, 31(3), 343–353. https://doi.org/10.1002/hpja.280

Rogers, R. W. (1975). A protection motivation theory of fear appeals and attitude change. The Journal of Psychology, 91(1), 93–114. https://doi.org/10.1080/00223980.1975.9915803

Witte, K. (1992). Putting the fear back into fear appeals: The extended parallel process model. Communication Monographs, 59(4), 329–349. https://doi.org/10.1080/03637759209376276</text>
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}}
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}

[[File:Woman receives mammogram (3).jpg|thumb|right|300px|Figure 1. A woman undergoing a standard mammogram procedure. Image by Rhoda Baer, National Cancer Institute, public domain.]]

Scenario

Megan, 50, discovers a letter in her mailbox inviting her a routine mammogram. She has continuously delayed it for the last four years. Her mother passed as a result of breast cancer, and the mere thought of the appointment makes her stomach knot. A part of her is comfortable with the uncertainty. The letter lays within a draw. In contrast, her friend Stella obtains the same letter a week later. Stella promptly schedules her appointment, Stella's view is synonymous with taking responsibility of her health. The procedure is the same, the same risk is attributed. Consequently, they feel completely differently about it and make opposite decisions.
{{RoundBoxBottom}}

Cancer screening programs for instance mammography, cervical screening, and bowl screening can significantly reduce mortality through early detection. Conversely, engagement remains below established national targets in Australia and internationally (Australian Institute of Health and Welfare, 2026). 

Key points this chapter will develop:
* Recognising these emotional dynamics attributes to screening programs to develop communication strategies that improve uptake.

* Individuals decision to participate in screening are not soley driven by rational evaluation; anxiety, fear, embarrassment, are all determinants as to whether individuals par take (Consedine et al., 2004)


{{RoundBoxTop|theme=2}}
'''Focus questions'''
* How can emotional determinants be addressed to improve screening participation? 
* What is the role of relief as an emotional driver in the uptake of screening?
* What emotions are experienced in relation to cancer screening?
* How does anxiety and fear act as barriers to screening uptake?
{{RoundBoxBottom}}

== What emotions are involved in cancer screening?==
* Emotional responses to screening comprise fear, anxiety, worry, stress, broadly conceptualised as anticipatory anxiety (Goodwin et al.,2023)
* Concern around the screening procedure and the cancer itself was reported between 10-50% of participants (Goodwin et al.,2023)
* A review of 74 studies identified that diagnostic anxiety tended to promote screening, conversely, procedural fear was almost reliably discouraging it (Goodwin et al., 2023).

==How does anxiety and fear motivate cancer screening?==
* Evidence from a systematic review illustrated that anxiety of a possible cancer diagnosis was often linked to greater screening uptake. Conversely, the fear of procedure discouraged participation (Goodwin et al.,2023).
* In a large Australian sample, screening engagement increased as an anxiety also increased but when anxiety was severe individuals were less likely to engage in screening (Anderson et al.,2023).
* Protection Motivation Theory proposes that individuals are highly probable to screen when they feel at harm and believe screening will benefit (Rogers, 1975).

==How does anxiety and fear act as barriers in cancer screening uptake?==
* Fear of receiving screening results can undermine participation, with fear of receiving a positive result as one of the biggest discouragers  of cervical screening among Australian women (Nagendiram et al., 2020).
* Severe anxiety may mitigate screening participation, with Australian literature illustrating lower bowel cancer screening participation across individuals experiencing severe anxiety (Anderson et al., 2023).
* Anticipatory anxiety associated around the screening procedure can attribute to avoidance, with never-screened or under screened Australian's noting psychological distress accompanied by invasive and sometimes discomfort-inducing nature of cervical screening (Creagh et al., 2021).

==Conclusion==
* The behavioural impacts of emotional responses on screening uptake, anxiety and fear proficient discouraging and motivating screening participation shaped by their focus and intensity
* Developing greater insight into the emotional components shaping screening decisions, guiding he development of strategies to mitigate psychological barriers promoting greater engagement
* Anxiety and fear influence screening in multifaceted ways, illustrating that emotional processes collectively shape screening behaviour.


==See also==
Anderson, L. E., Ireland, M. J., Myers, L., Avenell, C., Connaughton, T., &amp; Goodwin, B. C. (2023). Psychological distress and bowel cancer screening participation. Psycho-Oncology, 32(2), 229–236. https://doi.org/10.1002/pon.6072

Australian Institute of Health and Welfare. (2026). Cancer screening. Australian Government. https://www.aihw.gov.au/reports/australias-health/cancer-screening-and-treatment

Consedine, N. S., Magai, C., Krivoshekova, Y. S., Ryzewicz, L., &amp; Neugut, A. I. (2004). Fear, anxiety, worry, and breast cancer screening behavior: A critical review. Cancer Epidemiology, Biomarkers &amp; Prevention, 13(4), 501–510. https://doi.org/10.1158/1055-9965.501.13.4

Creagh, N. S., Zammit, C., Brotherton, J. M. L., Saville, M., McDermott, T., Nightingale, C., &amp; Kelaher, M. (2021). Self-collection cervical screening in the renewed National Cervical Screening Program: A qualitative study. Medical Journal of Australia, 215(8), 354–358. https://doi.org/10.5694/mja2.51137

Goodwin, B. C., Anderson, L., Collins, K., Sanjida, S., Riba, M., Singh, G. K., Campbell, K. M., Green, H., Ishaque, S., Kwok, A., Opozda, M. J., Pearn, A., Shaw, J., Sansom-Daly, U. M., Tsirgiotis, J. M., Janda, M., &amp; Grech, L. (2023). Anticipatory anxiety and participation in cancer screening: A systematic review. Psycho-Oncology, 32(12), 1773–1786. https://doi.org/10.1002/pon.6238

Nagendiram, A., Bougher, H., Banks, J., Hall, L., &amp; Heal, C. (2020). Australian women's self-perceived barriers to participation in cervical cancer screening: A systematic review. Health Promotion Journal of Australia, 31(3), 343–353. https://doi.org/10.1002/hpja.280

Rogers, R. W. (1975). A protection motivation theory of fear appeals and attitude change. The Journal of Psychology, 91(1), 93–114. https://doi.org/10.1080/00223980.1975.9915803</text>
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__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:Frustrated man at a desk (cropped).jpg|right|thumb|190px|'''Figure 1'''. Man frustrated at work due to lack of sleep]]

; Imagine this ... 
You wake up after several nights of poor sleep and immediately feel irritable and unmotivated. Nothing particularly bad has happened, yet this feeling follows you throughout the day. On your way to work, a minor delay in traffic feels unusually frustrating. At work, tasks that would normally seem manageable suddenly feel difficult and unrewarding. When a colleague gives you brief feedback, you interpret their comment more negatively than you normally would. By lunchtime, you avoid joining your coworkers because you do not feel like socialising. Later, a friend sends you a message asking if everything is okay, but you cannot identify a specific reason for feeling this way—you are simply &quot;in a bad mood.&quot;

The following day, after getting a good night's sleep and having an enjoyable conversation with a friend, the same workplace and responsibilities seem much easier to manage. You are more willing to talk to colleagues, everyday problems seem less frustrating, and activities feel more worthwhile.

Experiences like this demonstrate why mood is an important part of everyday life. Unlike an emotion that may be directed towards a specific event or person, a mood can persist across different situations and influence how we experience our surroundings. Why can the same workplace, people, and everyday challenges seem manageable one day but overwhelming the next? And how might our mood influence the way we think, behave, interact with others, and interpret what happens around us?
{{RoundBoxBottom}}

'''[[wikipedia:Mood_(psychology)|Mood]]''' is a long-lasting emotional stat that influences how a person experiences and responds to the world around them. Not to be confused with emotion which is directed towards a particular event or object. Mood affects emotions, thoughts and behaviours at the same time (Couch et al., 2005). Couch et al., 2005 also argues that there have been several attempts to explain the nature of mood and the best explanation for it is the [https://www.sciencedirect.com/topics/psychology/higher-order-cognition Higher Order Functional State Theory]. The theory argues that mood is a broader functional stat that influences emotional responses and behaviour, however, it is likely that that it may not completely capture subjective feeling of experiencing a mood (Couch et al., 2005)

Mood plays a crucial role in how a person experience, understand and interact with the world around them. Mood is not described as a feeling that lasts temporarily, such as happiness and sadness. Individuals are constantly experiencing a type of mood, even at a subconscious level. Mood plays an important role in how an individual find themselves situated within the world and how they experience their surroundings (Ratcliffe et al., 2013).

The importance of mood is that it helps influence the feelings of something meaningful or significant in life. Different people experience same environment differently based on their mood. Mood helps us establish a range of ways within which things are able to matter to us. This shows that mood does more than making an experience positive or negative, it creates possibilities that people recognise in their environment (Ratcliffe et al., 2013).

Mood disorders such as depression heavily affects how a person experiences the meaning of their surroundings. Activities that a person once participated in would suddenly seem less important or lose significance, and the person would no longer be able to experience emotional connect with others. In potential cases, some significances, such as feelings of threat, become overwhelming and influence many aspect of experience (Ratcliffe et al., 2013).

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What is Moodiness?
* What causes Moodiness?
* How can you manage Moodiness?

{{RoundBoxBottom}}

== What is Moodiness? ==
Mood is described as a long-lasting emotional state, which is developed by positive or negative experiences for an individual and how the individual will responds to similar experiences in the future. Instead of being a temporary emotional reaching, mood affect an individual’s sensitivity to potential rewards, punishment and threats across different situations and periods of time (Nettle et al., 2012).

The ability of mood can be experienced by many species like animals, this is due to the biological mechanisms involved in mood are similar across diverse animals (Nettle et al., 2012). Nettle et al., 2012 Argues that the ability of mood has been conserved through evolution and provides a key adaptive advantage.

Nettle et al., 2012 Explains that mood can be split into a two-dimensional framework. One of the dimensions explains a person’s sensitivity to punishment and threats, while the other dimension explains the sensitivity to reward.

Mood, instead of reacting to an event, influences the expectations of how certain of uncertain an experience is likely to be (Clark et al., 2018). An example of this is when you believe that your mood is affected by your bodily sensations, surrounding and potentially the consequence of your action. Mood disorders such as depression could be a causation of these predictions and uncertainty systems not functioning normally. (Clark et al., 2018). Clark et al., 2012 suggests that mood can be described as a background setting in the brain that influences how an individual interprets emotions, uncertainty and experiences.

'''Mood at a social level'''

A social understanding of mood can be described as something that influences how an individual interacts and behaves in a society. Social mood differs from emotion as it explains a larger reason for a what shapes an individual’s judgement, attitude and behaviours across different social situations (Squazzoni et al., 2017).

Squazzone et al., 2017 finds that mood can influence many areas of social life, such as relationships, consumer decisions, economic behaviour and voting preferences. Many times a decision that a person makes under the influence of their mood can seem irrational. Behavioural and evolutionary research implicates that mood can serve meaningful functions in guiding how an individual reacts to their social environment.

Overall, mood does not only work at an individual level. Shared and collective moods can have influence in social relationships and patterns of behaviours within groups and societies. It can be implied that mood playes an important role in shaping how an individual or a group perceive others, make decisions, form relationships and respond to wider social and political environments ( Squazzne et al., 2017)

'''Mood at a psychological level''' 

Within psychology, mood can be described as a recurring emotional state which influences a person’s thinking, feelings, evaluation of situations and behaviour. It differs from emotions which is more of an immediate response to a situation, mood can be highly subjective, where how a person experiences and describes their mood can be based on their own perceptions and personal experiences (Donnet et al., 1993). 

Mood has been long researched within the history of philosophy and medicine. Earlier examples were based off of the four humours – blood, phlegm, yellow bile and black bile, where mood would depend on balance between these bodily fluids. However, in modern psychology uses many other different approaches such as cognitive, phenomenological, psychoanalytic and ethological (Donnet et al., 1993).

To simplify things further, moods are like a lens we use to experience everyday life. For example, when a person is in a happy mood, everything in their surroundings and their interactions will appear to be more enjoyable and meaningful. In contrast, when someone is in a negative mood, the same situation all of a sudden will start to feel less appealing or significant (Freeman et al., 2014). Freeman et al., 2014 also argues that using psychology can be heavily beneficial to understand mood as not only something someone feels internally, but it can help shape how someone perceives, interprets and engages with their environment. 

'''Mood at a biological level'''

From the perspective of biology, mood can be influenced by body’s internal biological rhythms, mainly the [[wikipedia:Circadian_rhythm|circadian rhythm]], which is in charge of controlling the 24-hour sleep-wake cycle. This particular biological clock is managed mostly by the specific “clock” genes and is influenced by environmental signals, know as zeitgebers, such as exposure to light (Justice et al., 2006).

Mood disorders generally occur due to disruptions in these biological rhythms. An example of this is when changes in daylight occur due to seasonal change which affects mode and contributes to winter depression. The relationship between light, biological rhythms, and mood is supported by the effectiveness of light therapy, which helps in treating seasonal depression and potentially other forms of depression (Justice et al., 2006).

Hormone known as [[wikipedia:Melatonin|melatonin]] assists in regulating the circadian system and sleep, further demonstrating the connection between biological processes and mood (Justice et al., 2006). Any changes in the melatonin secretion and disrupted biological rhythms has been observed in people experiencing mood disorders (Mongeau et al., 2013). Research also suggests that genetic, epigenetic, and environmental factors can disrupt the alignment between a person’s internal body clock and external schedules, potentially increasing vulnerability to mood problems (Mongeau et al., 2013). It can be said in biological approach that mood is partly influenced by interactions with brain, hormones, sleep, genetics and the body’s internal clock. Any disruptions in these systems can result in changes in mood (Justice et al.,2006).

&lt;quiz display=simple&gt;
{Which statement best explains how mood influences an individual?
|type=&quot;()&quot;}
+ Mood acts like a background state that can influence how people interpret experiences, rewards, threats, and social situations.
- Mood is entirely determined by a person’s circadian rhythm.
- Mood only affects a person’s emotions and has no influence on behaviour or decision-making.
- Mood is a brief emotional reaction that disappears immediately after an event.

{Mood is influenced only by psychological experiences and is unrelated to biological processes such as sleep, circadian rhythms, hormones, and light exposure?
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==What Causes Moodiness?==

Changes in mood can be described as a combination of internal psychological processes, personality differences, and the situations that a person experiences. People are naturally able to regulate their moods by comparing how they currently vs how they would like to feel. The difference in actual vs preferred mood, caused people to use mood-regulation techniques to improve, maintain or on occasions worsen their mood (Wesniewska et al., 2017). 

The changes in mood can be caused by controlled or automatic process. In the controlled process, a person tries to consciously change how they feel, in comparison, in automatic process a person’s feelings change involuntarily. And example of this could be a person will automatically direct their attention towards an information that helps alter their current mood (Wesniewska et al., 2017).

Another huge factor in mood changes is a person’s personality. People that are extraverted are associated with having a positive affect while people with neuroticism are associated with having a negative affect. However, these can change based on the situation (Wesniewska et al., 2017).

Overall, changes in mood can be a result of interactions between personality, situations, cognitive process, and the strategies people use to regulate how they feel (Wesniewska et al., 2017)

'''Social Contributions to Moodiness'''

Social media and smartphones communication also play a big part in changes of mood in adolescents. Li et al., 2023 completed a study involving 83 adolescents over 90 days, analysing about 354,000 messages across social communication apps with participants providing ratings of their mood daily.

Li et al., 2023 found that the emotional tone of communications between adolescents were used as predictor of how they felt the following day. For example, adolescents that were involved in more positive language in their messages experienced more positive mood the next day, despite having a lower mood score the prior day. Larger used in positive language also caused reductions in levels of [[wikipedia:Anhedonia|anhedonia]], which is referred to as reduced pleasure or interest in activities. Alternatively greater use of negative language in communications resulted in increased [[wikipedia:Dysphoria|dysphoria]], or felling of emotional distress and unhappiness.

Li et al., 2023 also found that higher usage of socially engaging words such as words relating to friends and affiliation, predicted positive changes in mood. Frequent uses of emojis, specifically the heart emoji also associated with improved mood.

It can be argued that the findings find that there is a strong relationship in between social media communication and mood fluctuations. However the research primarily shows that online language is a good predictor for changes in mood fluctuations, instead of proving that social media communication causes mood changes (Li et al., 2023). 

'''How does childhood affect Moodiness'''

Childhood trauma can also play a major role in person developing mood disorders, some of which include depression and bipolar disorder. Trauma includes experiences such as physical, sexual or emotion abuse, neglect, parental loss or prolonged separation during childhood. Research shows that people with mood disorders are likely to share these experiences in comparison to the general population (Andryszewska et al., 2018).

One of the explanations could be that early trauma can have an affect on the developing brain. Stress during childhood years can release stress-related chemicals and neurotransmitters which can interfere with the developing brain cells and networks. This interference can have lasting affects on areas in charge of memory and emotional regulation, in particular the hippocampus and amygdala, potentially leading into having an affect on how an individual processes emotion and responds to stress later in life (Andrysweska et al., 2018).

Childhood trauma can also interact with genetic and epigenetic factors, potentially having an effect on how vulnerable a person is to future mood problems. HPA stress-response system and inflammation has also been associated with trauma which can have a further impact on mood disorders (Andrysweska et al., 2018).

It can be argued that childhood trauma does not necessarily cause mood disorder by itself. Instead, the research suggest that it can interact with brain development, stress systems, genetics, and epigenetic processes, which can all increase the risk and potentially the severity of mood disorders later in life (Andrysweska et al., 2018). 

&lt;quiz display=simple&gt;
{Which of the following best explains why a person’s mood may change?
|type=&quot;()&quot;}
- Mood changes are caused only by personality.
- Mood changes are caused only by social interactions.
+ Mood changes can result from interactions between personality, situations, cognitive processes, and mood-regulation strategies.
- Mood changes occur randomly and cannot be regulated.

{The use of positive language in adolescents’ online communication was associated with a more positive mood the following day?
|type=&quot;()&quot;}
+ True
- False
&lt;/quiz&gt;

==How Can You Manage Moodiness?==

Mood management plays a huge role in maintaining a healthy and prosperous live, managing and trying to actively maintain positive mood reduces negative mood throughout a person’s everyday behaviours and activities (Hess et al., 2010). Hess et al., argues that using the principle of homeostasis, where a person adjusts their behaviour to respond to a change in mood in order maintain a relatively stable emotional state. 

An example of this can be an individual seeking out an activity/hobby that helps them calm or raise positive mood when they might be experiencing a negative mood. Seeking these methods to maintain positive mood can result encouraging behaviours that assist in maintain that emotional state. Research found strong results for the mood management process, with positive mood typically lasting several hours (Hess et al., 2010)

Forgas et al., 2023 ran three experiments examining how people naturally regulate positive and negative moods over time. The participants first underwent mood inductions, which placed them either in a positive or negative mood. They were then asked to complete a sequence of tasks to observe if their responses reflected their initial mood or if it changed. It was found that people in positive mood responded positively while those in negative responded negatively. However, over time the responses moved away from their mood, which may be due to people trying to adjust their mood from to positive or negative. The experiment suggests that mood is not statice and people may use their cognitive process to move themselves into a balanced mood over time.

Overall, the process of mood management can be described as an ongoing process of regulating mood through behaviour. It is empirical that people seek activities/hobbies that influence their emotional state as it allows them to respond to changes in mood and maintain greater emotional stability in everyday life (Hess et al., 2010).

'''Participation in physical activity and its effect on mood'''

Physical activity plays a vital role in maintaining a positive effect on moon and mental well-being, specially for those that might be experiencing symptoms of depression and anxiety. Research has found that exercising regularly can help in reducing depressive symptoms. It can be argued that just one single session of exercise can temporarily improve depression and anxiety symptoms for up to several houre and in many cases, up to a day. People that participate in regular exercise report improvements in self-esteem, vitality, general well-being and satisfaction with their physical appearance (Peluso et al., 2005).

Frequently participating in both [[wikipedia:Aerobic_exercise|aerobic exercise]] such as walking, running or cycling and [[wikipedia:Anaerobic_exercise|anaerobic exercise]], like resistance training have been associated with reduction in depressive symptoms. However, it is found that the relationship between exercise and improved mood varies from individual to individual, specially in those without existing mental health symptoms (peluso et al., 2005).

There are many reasons for exercising improving mood, from a psychological perspective it could be distraction from negative thoughts or stressful situation while exercising. Self-efficacy which involves joys of completing a tedious physical activity resulting in confidence and feelings of achievement and social interactions where exercising with other provider social connection and support (Peluso et al., 2005).

Biological reasons could also result in positive moods where brain chemicals known as monoamines are involved in mood regulation. Exercising also increases the release of [[wikipedia:Endorphins|endorphins]] which contributes to feeling of calmness and improved mood after physical activity. However, research does not implicate the use of one biological mechanism as the sole explanation (Peluso et al., 2005)

Overall, research does suggest that participating in physical activity does have a positive effect on a person’s mood through a combination of psychological, social and biological processes. Exercise can help with reduction of negative feelings, assist with providing social connections and distraction, increase confidence and produce biological changes which are associated with improved emotional well-being (Peluso et al., 2005). 

'''Effect of music on mood'''

Suvi et al., 2007 researched the importance of music and how it helps adolescents manage and regulate moods in everyday life. Across three studies the researcher found that much plays an important role in emotional well being and adolescent development. It was very important for understanding, expressing, maintaining and changing moods also with supporting identity, independence and social relationships.

Suvi et al., 2007 found two main goals for using music to regulate mood: mood improvement and mood control. Mood improvement was described as a method to use music to create or strengthen positive feelings while mood control was described as using music to manage unwanted or difficult emotional states. Adolescents used seven different strategies to achieve these goals. '''Entertainment''' to help create and maintain positive moods, '''revival''' involved using music to increase energy and feel refreshed. '''Strong sensation''' which helped adolescents experience powerful emotions through music and '''Diversion''' which was used as a distraction from stress, negative thoughts and unwanted feelings.

'''Discharge''' was another strategy used which involved using music to let out anger and frustration, '''Mental work''' which was the use of music as a reflection and understanding personal feelings and experiences. Lastly, '''Solace''' which involved music as a comfort and emotional support during difficult situations (Suvi et al., 2007).

Suvi et al., 2007 developed a '''Music in Mood Regulation (MMR) scale''' which assisted in understanding how each adolescent used each strategy to manage moods. The results showed that girls relied a lot more on these strategies than boys while older adolescents used it more frequently than younger adolescents. It was also found that MMR was related to musical preferences, musical background and general mood-regulation abilities.

Overall, it was concluded that music can be a very valuable too in order to manage everyday mood. It can help with improving positive mood, help coping with negative moods, process feeling, find comfort and maintain control over emotional experiences. 

'''Use of online gaming to manage mood'''

Russoneillo et al.,2009 investigated the use of casual video games and if it can help manage people’s mood and stress. Casual video games are games that can be described as fun, easy to learn, quickly accessible and suitable for short periods of play. The test was a randomised controlled study in which the participants played one of three games: Bejewelled 2, bookworm adventures and peggle, for 20 minutes or completed and internet search task as a controlled condition. Mood was measured using the [[wikipedia:Profile_of_mood_states|Profile of Mood States (POMS)]], while [[wikipedia:Electroencephalography|EEG]] and heart-rate variability were also used to examine physiological changes.

The results found that playing casual video game helped provide an effective way of regulating negative moods and encouraging more positive emotional states. In between all three games, all participants showcased significant reductions in tension, with the games collectively reducing tension compared to the control condition. After playing all three games the participants too reported lower depressive mood. There were large reduction in depressive score for participants playing video games in comparison to those in the control group which was identified through the effect size analysis (Russoneillo et al., 2009).

Research also found that gaming also had reductions in anger, fatigue and confusion. However, the results did vary based on which game was played. The EEG supported the self-reported positive mood with brain activity associated with improved mood, greater engagement and emotional stability. Heart rate measure also suggested that certain game, particularly Bejeweled 2 produced physiological changes consistent with reduced physical stress (Russoneillo et al., 2009).

In summary, the study suggests that playing casual video games for short periods of time can have a positive effect on managing mood by reducing negative feelings such as tension, anger, depression, fatigue and confusion while promoting relaxation and positive engagement. However, it has been noted that the research did have methodological limitations and that further research is recommended before considering casual gaming as an established therapeutic intervention (Russoneillo et al., 2009). 

&lt;quiz display=simple&gt;
{Which of the following can be used as a strategy to help manage and improve mood?
|type=&quot;()&quot;}
- Physical activity.
- Listening to music.
- Playing casual video games.
+ All of the above.

{Mood is a completely fixed emotional state, meaning people cannot naturally regulate or change their mood over time?
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure; this can be the figure in the scenario
* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* Images must be embedded from [[commons:|Wikimedia Commons]] which hosts free-to-use media such as photos, diagrams, graphs, video, and audio
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Conclusion==
Mood is a complex and important psychological phenomenon that influences how individuals think, feel, behave, interpret their surroundings, and interact with other people. Rather than being a brief emotional reaction to a particular event, mood can persist across situations and influence sensitivity to rewards, threats, and future experiences (Nettle et al., 2012). Mood can therefore be viewed as a background state or “lens” through which people experience their everyday lives, affecting what appears meaningful, enjoyable, threatening, or worth pursuing (Ratcliffe et al., 2013; Freeman et al., 2014).

Importantly, no single explanation can fully account for why moods develop and change. From a biological perspective, circadian rhythms, sleep, melatonin, genetics, and other biological processes contribute to mood regulation, while disruptions to these systems have been associated with mood disorders (Justice et al., 2006; Mongeau et al., 2013). From a social perspective, mood can influence relationships, decision-making, social behaviour, and even wider economic and political preferences (Squazzoni et al., 2017). Individual differences are also important, as personality, cognitive processes, environmental situations, and automatic or controlled mood-regulation strategies can contribute to changes in mood (Wesniewska et al., 2017).

Understanding mood is particularly important because people can actively engage in behaviours that help manage and regulate their emotional states. Research suggests that music can help individuals improve, control, express, and process their moods (Suvi et al., 2007), while physical activity can improve mood through psychological, social, and biological mechanisms (Peluso et al., 2005). Casual video games may similarly reduce tension, anger, fatigue, confusion, and depressive mood while promoting relaxation and positive engagement (Russoniello et al., 2009).

Finally, understanding factors associated with negative mood is essential for mental health. Childhood trauma can interact with brain development, stress-response systems, genetic factors, and epigenetic processes, increasing vulnerability to later mood disorders (Andryszewska et al., 2018). Social communication may also provide insight into mood fluctuations, with positive online language predicting more positive subsequent mood among adolescents (Li et al., 2023). Overall, mood should be understood as the result of interacting biological, psychological, social, and environmental influences. Recognising these influences and developing healthy mood-management strategies may support emotional well-being and provide greater understanding of how everyday mood fluctuations can influence people's lives.

==See also==
* [[Motivation and emotion/Book/2025/Moodiness|Moodiness]] Honeybelle11 (Book chapter, 2025)
* [[Motivation and emotion/Book/2014/Mood and emotion|Mood and emotion]] (Book chapter, 2014)
* [[Motivation and emotion/Book/2020/Mood management theory and media consumption|Mood management theory and media consumption]] (Book chapter, 2020)

==References==
{{Hanging indent|1=
Nettle, D., &amp; Bateson, M. (2012). The Evolutionary Origins of Mood and Its Disorders. Current Biology, 22(17), R712–R721. Doi: https://doi.org/10.1016/j.cub.2012.06.020&lt;br&gt;

Clark, J. E., Watson, S., &amp; Friston, K. J. (2018). What is mood? A computational perspective. Psychological Medicine, 48(14), 2277–2284. doi: https://doi.org/10.1017/S0033291718000430&lt;br&gt;

Prescott-Couch, A. (2005). What is a Mood? The Yale Philosophy Review, 1, 42–58. Doi: https://doi.org/10.5840/ypr200513&lt;br&gt;
Europe PMC. (2019). Europe PMC. Europepmc.Org. doi: https://europepmc.org/article/med/8275897&lt;br&gt;

Freeman, L. (2014). Toward a Phenomenology of Mood. The Southern Journal of Philosophy, 52(4), 445–476. Doi: https://doi.org/10.1111/sjp.12089&lt;br&gt;

Wirz-Justice, A. (2006). Biological rhythm disturbances in mood disorders. International Clinical Psychopharmacology, 21, S11-S15. Doi: https://doi.org/10.1097/01.yic.0000195660.37267.cf&lt;br&gt;

Lanfumey, L., Mongeau, R., &amp; Hamon, M. (2013). Biological rhythms and melatonin in mood disorders and their treatments. Pharmacology &amp; Therapeutics, 138(2), 176–184. Doi: https://doi.org/10.1016/j.pharmthera.2013.01.005&lt;br&gt;

Doan, L. (2012). A Social Model of Persistent Mood States. Social Psychology Quarterly, 75(3), 198–218. Doi: https://doi.org/10.1177/0190272512451157&lt;br&gt;

Hess, J. D., Kacen, J. J. and Kim, J. (2006). Mood-management dynamics: The interrelationship between moods and behaviours. British Journal of Mathematical and Statistical Psychology, 59, 347–378. Doi: https://doi.org/10.1348/000711005X81133&lt;br&gt;

Forgas, J. P., &amp; Ciarrochi, J. V. (2002). On Managing Moods: Evidence for the Role of Homeostatic Cognitive Strategies in Affect Regulation. Personality and Social Psychology Bulletin, 28(3), 336–345. Doi: https://doi.org/10.1177/0146167202286005&lt;br&gt;

Peluso, M. A. M., &amp; Andrade, L. H. S. G. de. (2005). Physical activity and mental health: The association between exercise and mood. Clinics, 60(1), 61–70. Doi: https://doi.org/10.1590/S1807-59322005000100012&lt;br&gt;

Marszał-Wiśniewska, M., &amp; Nowicka, M. (2017). Individual Differences in Mood Changes. Journal of Happiness Studies, 19(5), 1415–1438. Doi: https://doi.org/10.1007/s10902-017-9879-5&lt;br&gt;

Jaworska-Andryszewska, P., &amp; Rybakowski, J. K. (2018). Childhood trauma in mood disorders: neurobiological mechanisms and implications for treatment. Pharmacological Reports, 71(1). Doi: https://doi.org/10.1016/j.pharep.2018.10.004&lt;br&gt;
Saarikallio, S. (2017). Music as mood regulation in adolescence. Jyväskylä Studies in Humanities, 67. Doi: https://doi.org/978-951-39-2731-8&lt;br&gt;

Li, L. Y., Trivedi, E., Helgren, F., Allison, G. O., Zhang, E., Buchanan, S. N., Pagliaccio, D., Durham, K., Allen, N. B., Auerbach, R. P., &amp; Shankman, S. A. (2023). Capturing mood dynamics through adolescent smartphone social communication. Journal of Psychopathology and Clinical Science, 132(8), 1072–1084. Doi: https://doi.org/10.1037/abn0000855&lt;br&gt;

Russoneillo, C., O’Brian, K., &amp; Parks, J. (2009). The effectiveness of casual video games in improving mood and decreasing stress (pp. 53–65). Journal of CyberTherapy &amp; Rehabilitation.
}}

==External links==
{{ic|Use bullet points as shown in Tutorial 2}}

[https://www.youtube.com/watch?v=E6Im__hq8Cc 10 Tips to help manage low moods | A doctor's guide] (Youtube)

[https://open.spotify.com/episode/3ZFpkCqQmUgWI1eQZf5MIF Managing mood swings &amp; emotional instability with Chrissy Court MSW, RSW] (Spotify)

[https://positivepsychology.com/positive-negative-emotions/ Positive and negative emotions: Do we need both?] (Positive psychology.com)

[https://www.mayoclinic.org/healthy-lifestyle/stress-management/in-depth/positive-thinking/art-20043950 Positive thinking: Stop negative self-talk to reduce stress] (Mayo clinic)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Mood]]</text>
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      <comment>brief descriptions on content and contributing</comment>
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      <text bytes="2139" sha1="6qn2yp5f3x9scoyjbwbgrrplmiy4odn" xml:space="preserve">{{mathematics}}

The goal of this Wikiversity course is to provide educational materials for pre-university mathematics. What is considered pre-university mathematics is not the same in every country, so some topics may be missing and some topics may be considered university level mathematics depending on where you live. The reader will have to use their best judgement on what content is relevant to their studies.

As with almost everything on Wikiversity, contributions to this course will be licensed by both the [https://creativecommons.org/licenses/by-sa/4.0/ CC-BY-SA 4.0 License] and the [https://www.gnu.org/copyleft/fdl.html GFDL]. A common exception is media stored on [https://commons.wikimedia.org Wikimedia Commons] and displayed on Wikiversity, which accepts more licenses. See [https://commons.wikimedia.org/wiki/Commons:Licensing#Well-known_licenses Commons:Licensing] for more details.

Contributing text that is considered your own original work is not too difficult in mathematics, but creating original graphics is both more difficult and time consuming. Please ensure all contributions are compatible if you choose to contribute.

The text is intentionally simple both to make language translation easier and to be more easily understood by readers not fluent in English. Please use short sentences, [https://en.wikipedia.org/wiki/Active_voice active voice], and simple vocabulary. Please avoid [https://en.wikipedia.org/wiki/Idiom idioms], [https://en.wikipedia.org/wiki/Slang slang], [https://en.wikipedia.org/wiki/Passive_voice passive voice], long sentences, complex sentences, 
[https://en.wikipedia.org/wiki/English_phrasal_verbs phrasal verbs], culture-specific references, ambiguous [https://en.wikipedia.org/wiki/Pronoun pronouns], and [https://en.wikipedia.org/wiki/Double_negative double negatives]. Use your best judgement on what is considered complex English based on the difficulty of the maths topic, such as simpler for a page on addition and more complex for a page on trigonometry.

===[[/Glossary/|Glossary]]===

===Algebra===
*[[/Algebra Functions/|Functions]]

===Geometry===

===Statistics===</text>
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      <text bytes="6270" sha1="eja2l4fmq77n8i6w5p6731108iip3qf" xml:space="preserve">{{mathematics}}

The goal of this Wikiversity course is to provide educational materials for pre-university mathematics. What is considered pre-university mathematics is not the same in every country, so some topics may be missing and some topics may be considered university level mathematics depending on where you live. The reader will have to use their best judgement on what content is relevant to their studies.

Please see [[Pre-University Mathematics#Contributing|Contributing]] below for compatible content rules and creating accessible content.

===[[/Glossary/|Glossary]]===

===Algebra===
*[[/Algebra Functions/|Functions]]

===Geometry===

===Statistics===

== Contributing ==

As with almost everything on Wikiversity, contributions to this course will be licensed by both the [https://creativecommons.org/licenses/by-sa/4.0/ CC-BY-SA 4.0 License] and the [https://www.gnu.org/copyleft/fdl.html GFDL]. A common exception is media stored on [https://commons.wikimedia.org Wikimedia Commons] and displayed on Wikiversity, which accepts more licenses. See [https://commons.wikimedia.org/wiki/Commons:Licensing#Well-known_licenses Commons:Licensing] for more details.

Contributing text that is considered your own original work is not too difficult in mathematics, but creating original graphics is both more difficult and time consuming. Please ensure all contributions are compatible if you choose to contribute.

=== Write for English learners ===

The text is intentionally simple both to make language translation easier and to be more easily understood by readers not fluent in English. Please use short sentences, [https://en.wikipedia.org/wiki/Active_voice active voice], and simple vocabulary. Please avoid [https://en.wikipedia.org/wiki/Idiom idioms], [https://en.wikipedia.org/wiki/Slang slang], [https://en.wikipedia.org/wiki/Passive_voice passive voice], long sentences, complex sentences, 
[https://en.wikipedia.org/wiki/English_phrasal_verbs phrasal verbs], culture-specific references, ambiguous [https://en.wikipedia.org/wiki/Pronoun pronouns], and [https://en.wikipedia.org/wiki/Double_negative double negatives]. Use your best judgement on what is considered complex English based on the difficulty of the maths topic, such as simpler for a page on addition and more complex for a page on trigonometry.

=== Write for an International Audience ===

Examples
* SI units instead of the imperial system.
* Aluminium instead of aluminum.
* Maths instead of math.
* Exercises instead of problems.

=== Write for the Student but also for the Educator ===

Educational text often has an underlying pattern that guides the student but is not explicitly written. An example would be students learning how to solve a linear equation in one variable. 
* Students first use the idea of a two-pan balance scale with identical but unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, finding the weight of an &lt;math&gt;x&lt;/math&gt; by adding and removing weights from the scale so that the scale remains balanced. (Provides an accessible analogy to students before transitioning to equations.)
* Students redo the same two-pan balance scale exercises in the form of equations.
* The first set of exercises require only addition and subtraction, will have small and positive integers in every step to solve the equation, and the solution is a positive integer. (These problems are intentionally simple and students may try to guess the solution.)
* Multiplication and division are required to solve the next set of exercises and the solution is still a positive integer.
* The next set of exercises will have positive fractions as the solution, fractions when solving, or both. (Students can no longer reliably guess the solution.)
* The next set of exercises will have positive and negative integers and fractions as the solution. (Helpful analogies, like an equation as a two-pan balance scale with unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, do not work well when the unknown weights &lt;math&gt;x&lt;/math&gt; have negative mass and the solution process become more abstract.)
* Equations that have zero solutions and infinite solutions are introduced. (A linear equation can have &lt;math&gt;0&lt;/math&gt;, &lt;math&gt;1&lt;/math&gt;, or infinite solutions. In the case of &lt;math&gt;0&lt;/math&gt; solutions, the equation simplifies to something like &lt;math&gt;4 = 9&lt;/math&gt;, which is not possible and no value of &lt;math&gt;x&lt;/math&gt; will solve the equation. In the case of infinite solutions, the equation starts out with identical left and right hand sides, such as &lt;math&gt;4x-3 = 4x-3&lt;/math&gt;, which will work no mater what &lt;math&gt;x&lt;/math&gt; is.)

The text in parentheses is for the educator and explicitly states why the exercises are in the order they are. The educator could be a classroom teacher or a family member of the student. The classroom educator will likely understand the reasoning of the exercises and their ordering without the text in parentheses. A family member may have learned to solve a linear equation many years ago or never learned. The text in parentheses is extremely helpful to this less qualified educator.

Please include notes to the educator like in the above example. The notes explicitly state the design decisions, how understanding is evolving in the mind of the student, and any other information that the educator can use to better teach the student.

=== Write Mathematical Symbols and Equations in TeX ===

TeX renders maths symbols in a much nicer format than regular text using &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; in the page source. Wikiversity has a page on [[Help:Formula|writing in TeX]] with many examples.

The source,

&lt;code&gt;&lt;nowiki&gt;
&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
&lt;/nowiki&gt;&lt;/code&gt;

is rendered as,

&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
 
The enclosing &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;center&quot;&gt; ... &lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt; is for centering the rendered equation. The &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; encloses the TeX markup language. 

=== Creating New Pages ===

Please do not add blank pages and expect someone else to finish them. Having incomplete pages makes the entire course look incomplete. There is a [[/Wishlist/|Wishlist]] page if you must.</text>
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==Topic development feedback==
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The [[Motivation and emotion/Assessment/Topic|topic development]] has been reviewed according to the [[Motivation and emotion/Assessment/Topic#Marking criteria|marking criteria]]. Written feedback is provided below, plus see the [[Motivation and emotion/Assessment/Topic/Feedback|general feedback]] page. Also check the [[Special:History/{{PAGENAME}}|page history]] for changes made whilst reviewing the plan. If you don't understand the feedback or would like further information, [[Motivation and emotion/Staff|get in touch]] to discuss. Marks are available via {{Motivation and emotion/Canvas}}. Marks are based on the latest version before the due date.
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{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
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#
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&lt;!-- References --&gt;
#
|8=
&lt;!-- Resources --&gt;
#
|9=
&lt;!-- User page --&gt;
#
|10=
&lt;!-- Social contribution --&gt;
#
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:42, 17 August 2025 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template which includes commonly used feedback comments:
&lt;pre&gt;
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
# Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed)
# User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]
# Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement
# Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand)
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
# Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings)
# Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings)
&lt;!-- 3-level --&gt;
# Overly complicated[[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying
&lt;!-- Conceptual --&gt;
# Messy heading structure – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# The headings lack sufficient incision into, and exposition of, the topic
# Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts.
&lt;!-- Other ---&gt;
# Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections
# The Overview and Conclusion should not use sub-headings
# Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.)
# Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings
# &quot;Introduction&quot; heading isn't necessary – provide this information in Overview or move into subsequent sections
# Cover definition(s) in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information
# Case study doesn't need a separate heading; instead embed case study within relevant sections
# Quiz doesn't need a separate heading; instead embed quiz questions within relevant sections
# Check grammar (e.g,. missing question mark)
# Remove [[wikt:acronym#Noun|acronym]]s from headings
# Remove citations from headings
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure
# Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement
# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
# Reasonably good alignment between focus questions and heading structure, but aim for closer alignment
# Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve.
# Develop closer alignment between sub-title, focus questions, and top-level headings
# Insufficient alignment between sub-title, focus questions, and top-level headings
&lt;!-- GenAI ---&gt;
# Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
# Very good
# Good
# Basic
# Insufficient
# Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- GenAI ---&gt;
# Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
# A scenario or case study is presented in a feature box at the start of this section; I moved an image into the feature box to help attract reader interest
# A scenario or case study is presented in a feature box at the start of this section; add an image to the scenario to help attract reader interest
# Move the scenario or case study into a feature box (with an image) to the start of this section to help engage reader interest
# A scenario or case study is planned
# Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
# A promising description of the problem/topic is planned or presented
# A basic description of the problem/topic is planned or presented
# Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections.
# Add a brief, evocative description of the problem/topic
&lt;!-- Style --&gt;
# Use present, rather than future, tense
# Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios)
&lt;!-- Focus questions --&gt;
# Focus questions are aligned with sub-title and top-level headings
# Reasonably good alignment between focus questions and heading structure, but consider closer alignment
# Develop closer alignment between the sub-title, focus questions, and top-level headings
# Use open- rather then close-ended focus questions
# Use single- rather than double-barrelled focus questions
# Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Present focus questions in a feature box at the end of this section
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Excellent – key points are well developed for each section
# Solid development
# Promising development
# Highlight the most relevant theories and synthesise the best research on the topic
# Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content
# Basic development
# Partial development
# Insufficient development
# No development
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
# The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title
# It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title
# It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
# All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
&lt;!-- Writing style --&gt;
# The writing style is clear and easy to follow
# The writing style is generally clear but could be simplified or made more concise
# The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.)
&lt;!-- Theory and research --&gt;
# Good balance of theory and research
# Promising balance of theory and research
# Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research
# Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. 
# Select the best theories about this topic
# Select the best research about this topic
&lt;!-- Citations --&gt;
# Excellent use of citations
# Very good use of citations
# Good use of citations
# Promising use of citations
# Basic use of citations
# Insufficient use of citations
# Non-peer-reviewed sources should be moved to the &quot;External links&quot; section
# Tip: Rather than starting with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence
&lt;!-- Citation style --&gt;
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials)
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year)
# [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min)
&lt;!-- Other --&gt;
# For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings
# ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title.
# Direct quotes need page numbers (APA style) – even better, express the idea in your own words
# Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a &quot;down&quot; style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info]
# Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective
# Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour)
# Move references into the References section. Keep citations in the main body.
# Consider using the [https://unicanberra.instructure.com/courses/15707/external_tools/262?display=borderless Studiosity] service and/or a service like [https://www.grammarly.com/ Grammarly] to help improve the quality of written expression and to check grammatical and spelling errors
&lt;!-- GenAI ---&gt;
# Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]].
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is well developed
# Conclusion is well underway
# Conclusion is underway
# Conclusion is underdeveloped
# Conclusion hasn't been developed
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Excellent - Relevant figure(s) presented, captioned, and cited
# Relevant figure(s) are presented and captioned
# Relevant figure(s) are presented
# The relevance of the figure to the topic is unclear
# A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Caption --&gt;
# Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text
# Figure caption(s) provide(s) a reasonably clear description that is connected with the main text
# Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved
# Figure caption(s) could better explain how the image connects to key points being made in the main text
# Figure caption(s) should include '''Figure X'''. ...
&lt;!-- Cite --&gt;
# Figure(s) are cited at least once in the main text
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
&lt;!-- Size --&gt;
# Consider increasing image size(s) (especially if they have text) to make them easier to view
# Consider decreasing image size(s) to make them less dominant
&lt;!-- Creation --&gt;
# Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Excellent use of scenarios/examples/case studies
# Promising use of scenarios/examples/case studies
# Keep scenarios brief 
# Basic use of scenario/example/case study
# Placeholder use of scenarios/examples/case studies
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
# Excellent use of quiz question(s)
# Promising use of quiz question(s)
# Placeholder use of quiz question(s)
# Place quiz each question in the most relevant section
# Focus the quiz question(s) on the take-home messages
# Consider including quiz question(s) about the take-home messages
&lt;!-- Tables --&gt;
# Excellent use of table(s)
# Promising use of table(s)
# Use APA style for table captions
# Add table caption
# Cite each table at least once in the text
# Include citations for sources of information presented in the table
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Excellent
# Very good
# Good
# Basic
# Insufficient
# To be developed
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
# At least one relevant systematic review and/or meta-analysis has been identified
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- Move --&gt;
# Move Wikipedia links to the &quot;See also&quot; section
# Move non-academic / non-peer reviewed sources to the &quot;External links&quot; section
&lt;!-- Citations --&gt;
# All references need in-text citation
# All citations need to be in the References
# Only include references which have been accessed and read
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## alphabetical order
## capitalisation
## [[Help:Wikitext quick reference|italicisation]]
## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting]
## make doi hyperlinks active (i.e., clickable)
## use dois where available instead of other links
## include hyperlinked dois
## page numbers should be separated by an en-dash (–) rather than a hyphen (-)
# A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic
# Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used
# Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity.
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
## Very good
## Good
## Basic
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## One of two link types provided
### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]]
### Also link to relevant [[w:|Wikipedia]] pages
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters)
## Use alphabetical order
&lt;!-- External links --&gt;
# External links
## Excellent
## Very good
## Good
## Basic
## One of two required external links provided
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Move Wikipedia link(s) to the &quot;See also&quot; section
## Move academic sources into the &quot;References&quot; sections and provide in-text citation
## Only include links directly related to the sub-title
## Target an international audience; Australians only represent 0.33% of the world population
## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link
## Use alphabetical order
## Link to the most relevant external resources about this topic
|9=
&lt;!-- User page --&gt;
# Excellent
# Used effectively
# Very good
# Good
# Basic
# Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
&lt;!-- Description about self --&gt;
# Excellent description about self provided
# Description about self provided
# Brief description about self – consider expanding
# Very brief description about self – consider expanding
# Add description about self
&lt;!-- Links to profile(s) --&gt;
# Link(s) provided to professional profile(s)
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
# Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Add link to book chapter
|10=
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
# Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making:
# One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
#* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]]
#* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum
# None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]].
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
# Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity.
# Well done on creating and uploading your own image!
# Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Descriptions of contributions could be more precise/accurate/detailed
# Add a brief summary of each contribution
# Remember to sign comments on talk pages
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
# Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed)
# User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]
# Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement
# Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand)
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
# Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings)
# Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings)
&lt;!-- 3-level --&gt;
# Overly complicated [[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying
&lt;!-- Conceptual --&gt;
# Messy heading structure – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# The headings lack sufficient incision into, and exposition of, the topic
# Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts.
&lt;!-- Other ---&gt;
# Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections
# The Overview and Conclusion should not use sub-headings
# Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.)
# Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings
# &quot;Introduction&quot; heading isn't necessary – provide this information in Overview or move into subsequent sections
# Cover definition(s) in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information
# Case study doesn't need a separate heading; instead embed case study within relevant sections
# Quiz doesn't need a separate heading; instead embed quiz questions within relevant sections
# Check grammar (e.g,. missing question mark)
# Remove [[wikt:acronym#Noun|acronym]]s from headings
# Remove citations from headings
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure
# Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement
# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
# Reasonably good alignment between focus questions and heading structure, but aim for closer alignment
# Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve.
# Develop closer alignment between sub-title, focus questions, and top-level headings
# Insufficient alignment between sub-title, focus questions, and top-level headings
&lt;!-- GenAI ---&gt;
# Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
# Very good
# Good
# Basic
# Insufficient
# Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- GenAI ---&gt;
# Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
# A scenario or case study is presented in a feature box at the start of this section; I moved an image into the feature box to help attract reader interest
# A scenario or case study is presented in a feature box at the start of this section; add an image to the scenario to help attract reader interest
# Move the scenario or case study into a feature box (with an image) to the start of this section to help engage reader interest
# A scenario or case study is planned
# Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
# A promising description of the problem/topic is planned or presented
# A basic description of the problem/topic is planned or presented
# Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections.
# Add a brief, evocative description of the problem/topic
&lt;!-- Style --&gt;
# Use present, rather than future, tense
# Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios)
&lt;!-- Focus questions --&gt;
# Focus questions are aligned with sub-title and top-level headings
# Reasonably good alignment between focus questions and heading structure, but consider closer alignment
# Develop closer alignment between the sub-title, focus questions, and top-level headings
# Use open- rather then close-ended focus questions
# Use single- rather than double-barrelled focus questions
# Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Present focus questions in a feature box at the end of this section
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Excellent – key points are well developed for each section
# Solid development
# Promising development
# Highlight the most relevant theories and synthesise the best research on the topic
# Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content
# Basic development
# Partial development
# Insufficient development
# No development
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
# The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title
# It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title
# It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
# All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
&lt;!-- Writing style --&gt;
# The writing style is clear and easy to follow
# The writing style is generally clear but could be simplified or made more concise
# The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.)
&lt;!-- Theory and research --&gt;
# Good balance of theory and research
# Promising balance of theory and research
# Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research
# Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. 
# Select the best theories about this topic
# Select the best research about this topic
&lt;!-- Citations --&gt;
# Excellent use of citations
# Very good use of citations
# Good use of citations
# Promising use of citations
# Basic use of citations
# Insufficient use of citations
# Non-peer-reviewed sources should be moved to the &quot;External links&quot; section
# Tip: Rather than starting with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence
&lt;!-- Citation style --&gt;
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials)
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year)
# [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min)
&lt;!-- Other --&gt;
# For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings
# ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title.
# Direct quotes need page numbers (APA style) – even better, express the idea in your own words
# Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a &quot;down&quot; style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info]
# Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective
# Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour)
# Move references into the References section. Keep citations in the main body.
# Consider using the [https://unicanberra.instructure.com/courses/15707/external_tools/262?display=borderless Studiosity] service and/or a service like [https://www.grammarly.com/ Grammarly] to help improve the quality of written expression and to check grammatical and spelling errors
&lt;!-- GenAI ---&gt;
# Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]].
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is well developed
# Conclusion is well underway
# Conclusion is underway
# Conclusion is underdeveloped
# Conclusion hasn't been developed
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Excellent - Relevant figure(s) presented, captioned, and cited
# Relevant figure(s) are presented and captioned
# Relevant figure(s) are presented
# The relevance of the figure to the topic is unclear
# A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Caption --&gt;
# Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text
# Figure caption(s) provide(s) a reasonably clear description that is connected with the main text
# Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved
# Figure caption(s) could better explain how the image connects to key points being made in the main text
# Figure caption(s) should include '''Figure X'''. ...
&lt;!-- Cite --&gt;
# Figure(s) are cited at least once in the main text
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
&lt;!-- Size --&gt;
# Consider increasing image size(s) (especially if they have text) to make them easier to view
# Consider decreasing image size(s) to make them less dominant
&lt;!-- Creation --&gt;
# Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Excellent use of scenarios/examples/case studies
# Promising use of scenarios/examples/case studies
# Keep scenarios brief 
# Basic use of scenario/example/case study
# Placeholder use of scenarios/examples/case studies
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
# Excellent use of quiz question(s)
# Promising use of quiz question(s)
# Placeholder use of quiz question(s)
# Place quiz each question in the most relevant section
# Focus the quiz question(s) on the take-home messages
# Consider including quiz question(s) about the take-home messages
&lt;!-- Tables --&gt;
# Excellent use of table(s)
# Promising use of table(s)
# Use APA style for table captions
# Add table caption
# Cite each table at least once in the text
# Include citations for sources of information presented in the table
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Excellent
# Very good
# Good
# Basic
# Insufficient
# To be developed
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
# At least one relevant systematic review and/or meta-analysis has been identified
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- Move --&gt;
# Move Wikipedia links to the &quot;See also&quot; section
# Move non-academic / non-peer reviewed sources to the &quot;External links&quot; section
&lt;!-- Citations --&gt;
# All references need in-text citation
# All citations need to be in the References
# Only include references which have been accessed and read
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## alphabetical order
## capitalisation
## [[Help:Wikitext quick reference|italicisation]]
## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting]
## make doi hyperlinks active (i.e., clickable)
## use dois where available instead of other links
## include hyperlinked dois
## page numbers should be separated by an en-dash (–) rather than a hyphen (-)
# A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic
# Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used
# Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity.
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
## Very good
## Good
## Basic
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## One of two link types provided
### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]]
### Also link to relevant [[w:|Wikipedia]] pages
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters)
## Use alphabetical order
&lt;!-- External links --&gt;
# External links
## Excellent
## Very good
## Good
## Basic
## One of two required external links provided
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Move Wikipedia link(s) to the &quot;See also&quot; section
## Move academic sources into the &quot;References&quot; sections and provide in-text citation
## Only include links directly related to the sub-title
## Target an international audience; Australians only represent 0.33% of the world population
## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link
## Use alphabetical order
## Link to the most relevant external resources about this topic
|9=
&lt;!-- User page --&gt;
# Excellent
# Used effectively
# Very good
# Good
# Basic
# Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
&lt;!-- Description about self --&gt;
# Excellent description about self provided
# Description about self provided
# Brief description about self – consider expanding
# Very brief description about self – consider expanding
# Add description about self
&lt;!-- Links to profile(s) --&gt;
# Link(s) provided to professional profile(s)
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
# Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Add link to book chapter
|10=
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
# Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making:
# One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
#* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]]
#* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum
# None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]].
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
# Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity.
# Well done on creating and uploading your own image!
# Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Descriptions of contributions could be more precise/accurate/detailed
# Add a brief summary of each contribution
# Remember to sign comments on talk pages
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:58, 26 August 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
* [[Template:MEBF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Topic]]
&lt;/noinclude&gt;</text>
      <sha1>5grfvyoyov5xr5zgikcguia7a60zkx3</sha1>
    </revision>
    <revision>
      <id>2829840</id>
      <parentid>2829832</parentid>
      <timestamp>2026-08-31T07:15:07Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples.</comment>
      <origin>2829840</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="47523" sha1="0puidow9nvr82dkhk07m60y1tjscp9j" xml:space="preserve">&lt;noinclude&gt;{{note|Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise for [[motivation and emotion]].&lt;br&gt;&lt;br&gt;[[Help:Transclusion|Transclude]] on a chapter [[Help:Talk page|talk page]].}}
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==Topic development feedback==
{{RoundBoxTop|theme=8}}
The [[Motivation and emotion/Assessment/Topic|topic development]] has been reviewed according to the [[Motivation and emotion/Assessment/Topic#Marking criteria|marking criteria]]. Written feedback is provided below, plus see the [[Motivation and emotion/Assessment/Topic/Feedback|general feedback]] page. Also check the [[Special:History/{{PAGENAME}}|page history]] for changes made whilst reviewing the plan. If you don't understand the feedback or would like further information, [[Motivation and emotion/Staff|get in touch]] to discuss. Marks are available via {{Motivation and emotion/Canvas}}. Marks are based on the latest version before the due date.
{{RoundBoxBottom}}
{{RoundBoxTop|theme=9}}
[[File:Autoroute icone.svg|right|85px]]
===1. [[Motivation and emotion/Assessment/Topic#Title|Title]]===
{{{1|No comment}}}
===2. [[Motivation and emotion/Assessment/Topic#Headings|Headings]]===
{{{2|No comment}}}
===3. [[Motivation and emotion/Assessment/Topic#Headings|Overview]]===
{{{3|No comment}}}
===4. [[Motivation and emotion/Assessment/Topic#Key points|Key points]]===
{{{4|No comment}}}
===5. [[Motivation and emotion/Assessment/Topic#Figure|Figure]]===
{{{5|No comment}}}
===6. [[Motivation and emotion/Assessment/Topic#Learning feature|Learning feature]]===
{{{6|No comment}}}
===7. [[Motivation and emotion/Assessment/Topic#References|References]]===
{{{7|No comment}}}
===8. [[Motivation and emotion/Assessment/Topic#Resources|Resources]]===
{{{8|No comment}}}
===9. [[Motivation and emotion/Assessment/Topic#User page|User page]]===
{{{9|No comment}}}
===10. [[Motivation and emotion/Assessment/Topic#Social contribution|Social contribution]]===
{{{10|No comment}}}
{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
#
|2=
&lt;!-- Headings --&gt;
#
|3=
&lt;!-- Overview --&gt;
#
|4=
&lt;!-- Key points--&gt;
#
|5=
&lt;!-- Figure --&gt;
#
|6=
&lt;!-- Learning feature --&gt;
#
|7=
&lt;!-- References --&gt;
#
|8=
&lt;!-- Resources --&gt;
#
|9=
&lt;!-- User page --&gt;
#
|10=
&lt;!-- Social contribution --&gt;
#
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
#
|2=
&lt;!-- Headings --&gt;
#|3=
&lt;!-- Overview --&gt;
#
|4=
&lt;!-- Key points--&gt;
#
|5=
&lt;!-- Figure --&gt;
#
|6=
&lt;!-- Learning feature --&gt;
#
|7=
&lt;!-- References --&gt;
#
|8=
&lt;!-- Resources --&gt;
#
|9=
&lt;!-- User page --&gt;
#
|10=
&lt;!-- Social contribution --&gt;
#
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:42, 17 August 2025 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template which includes commonly used feedback comments:
&lt;pre&gt;
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
# Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed)
# User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]
# Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement
# Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand)
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
# Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings)
# Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings)
&lt;!-- 3-level --&gt;
# Overly complicated[[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying
&lt;!-- Conceptual --&gt;
# Messy heading structure – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# The headings lack sufficient incision into, and exposition of, the topic
# Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts.
&lt;!-- Other ---&gt;
# Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections
# The Overview and Conclusion should not use sub-headings
# Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.)
# Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings
# &quot;Introduction&quot; heading isn't necessary – provide this information in Overview or move into subsequent sections
# Cover definition(s) in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information
# Case study doesn't need a separate heading; instead embed case study within relevant sections
# Quiz doesn't need a separate heading; instead embed quiz questions within relevant sections
# Check grammar (e.g,. missing question mark)
# Remove [[wikt:acronym#Noun|acronym]]s from headings
# Remove citations from headings
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure
# Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement
# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
# Reasonably good alignment between focus questions and heading structure, but aim for closer alignment
# Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve.
# Develop closer alignment between sub-title, focus questions, and top-level headings
# Insufficient alignment between sub-title, focus questions, and top-level headings
&lt;!-- GenAI ---&gt;
# Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
# Very good
# Good
# Basic
# Insufficient
# Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- GenAI ---&gt;
# Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
# A scenario or case study is presented in a feature box at the start of this section; I moved an image into the feature box to help attract reader interest
# A scenario or case study is presented in a feature box at the start of this section; add an image to the scenario to help attract reader interest
# Move the scenario or case study into a feature box (with an image) to the start of this section to help engage reader interest
# A scenario or case study is planned
# Make the relevance of the scenario to the topic more clear
# Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
# A promising description of the problem/topic is planned or presented
# A basic description of the problem/topic is planned or presented
# Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections.
# Add a brief, evocative description of the problem/topic
&lt;!-- Style --&gt;
# Use present, rather than future, tense
# Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios)
&lt;!-- Focus questions --&gt;
# Focus questions are aligned with sub-title and top-level headings
# Reasonably good alignment between focus questions and heading structure, but consider closer alignment
# Develop closer alignment between the sub-title, focus questions, and top-level headings
# Use open- rather then close-ended focus questions
# Use single- rather than double-barrelled focus questions
# Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Present focus questions in a feature box at the end of this section
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Excellent – key points are well developed for each section
# Solid development
# Promising development
# Highlight the most relevant theories and synthesise the best research on the topic
# Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content
# Basic development
# Partial development
# Insufficient development
# No development
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
# The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title
# It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title
# It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
# All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
&lt;!-- Writing style --&gt;
# The writing style is clear and easy to follow
# The writing style is generally clear but could be simplified or made more concise
# The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.)
&lt;!-- Theory and research --&gt;
# Good balance of theory and research
# Promising balance of theory and research
# Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research
# Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples.
# Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples.
# Select the best theories about this topic
# Select the best research about this topic
&lt;!-- Citations --&gt;
# Excellent use of citations
# Very good use of citations
# Good use of citations
# Promising use of citations
# Basic use of citations
# Insufficient use of citations
# Non-peer-reviewed sources should be moved to the &quot;External links&quot; section
# Tip: Rather than starting with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence
&lt;!-- Citation style --&gt;
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials)
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year)
# [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min)
&lt;!-- Other --&gt;
# For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings
# ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title.
# Direct quotes need page numbers (APA style) – even better, express the idea in your own words
# Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a &quot;down&quot; style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info]
# Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective
# Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour)
# Move references into the References section. Keep citations in the main body.
# Consider using the [https://unicanberra.instructure.com/courses/15707/external_tools/262?display=borderless Studiosity] service and/or a service like [https://www.grammarly.com/ Grammarly] to help improve the quality of written expression and to check grammatical and spelling errors
&lt;!-- GenAI ---&gt;
# Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]].
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is well developed
# Conclusion is well underway
# Conclusion is underway
# Conclusion is underdeveloped
# Conclusion hasn't been developed
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Excellent - Relevant figure(s) presented, captioned, and cited
# Relevant figure(s) are presented and captioned
# Relevant figure(s) are presented
# The relevance of the figure to the topic is unclear
# A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Caption --&gt;
# Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text
# Figure caption(s) provide(s) a reasonably clear description that is connected with the main text
# Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved
# Figure caption(s) could better explain how the image connects to key points being made in the main text
# Figure caption(s) should include '''Figure X'''. ...
&lt;!-- Cite --&gt;
# Figure(s) are cited at least once in the main text
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
&lt;!-- Size --&gt;
# Consider increasing image size(s) (especially if they have text) to make them easier to view
# Consider decreasing image size(s) to make them less dominant
&lt;!-- Creation --&gt;
# Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Excellent use of scenarios/examples/case studies
# Promising use of scenarios/examples/case studies
# Keep scenarios brief 
# Basic use of scenario/example/case study
# Placeholder use of scenarios/examples/case studies
# Make the relevance of the scenario to the topic more clear
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
# Excellent use of quiz question(s)
# Promising use of quiz question(s)
# Placeholder use of quiz question(s)
# Place quiz each question in the most relevant section
# Focus the quiz question(s) on the take-home messages
# Consider including quiz question(s) about the take-home messages
&lt;!-- Tables --&gt;
# Excellent use of table(s)
# Promising use of table(s)
# Use APA style for table captions
# Add table caption
# Cite each table at least once in the text
# Include citations for sources of information presented in the table
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Excellent
# Very good
# Good
# Basic
# Insufficient
# To be developed
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
# At least one relevant systematic review and/or meta-analysis has been identified
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- Move --&gt;
# Move Wikipedia links to the &quot;See also&quot; section
# Move non-academic / non-peer reviewed sources to the &quot;External links&quot; section
&lt;!-- Citations --&gt;
# All references need in-text citation
# All citations need to be in the References
# Only include references which have been accessed and read
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## alphabetical order
## capitalisation
## [[Help:Wikitext quick reference|italicisation]]
## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting]
## make doi hyperlinks active (i.e., clickable)
## use dois where available instead of other links
## include hyperlinked dois
## page numbers should be separated by an en-dash (–) rather than a hyphen (-)
# A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic
# Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used
# Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity.
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
## Very good
## Good
## Basic
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## One of two link types provided
### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]]
### Also link to relevant [[w:|Wikipedia]] pages
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters)
## Use alphabetical order
&lt;!-- External links --&gt;
# External links
## Excellent
## Very good
## Good
## Basic
## One of two required external links provided
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Move Wikipedia link(s) to the &quot;See also&quot; section
## Move academic sources into the &quot;References&quot; sections and provide in-text citation
## Only include links directly related to the sub-title
## Target an international audience; Australians only represent 0.33% of the world population
## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link
## Use alphabetical order
## Link to the most relevant external resources about this topic
|9=
&lt;!-- User page --&gt;
# Excellent
# Used effectively
# Very good
# Good
# Basic
# Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
&lt;!-- Description about self --&gt;
# Excellent description about self provided
# Description about self provided
# Brief description about self – consider expanding
# Very brief description about self – consider expanding
# Add description about self
&lt;!-- Links to profile(s) --&gt;
# Link(s) provided to professional profile(s)
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
# Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Add link to book chapter
|10=
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
# Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making:
# One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
#* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]]
#* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum
# None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]].
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
# Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity.
# Well done on creating and uploading your own image!
# Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Descriptions of contributions could be more precise/accurate/detailed
# Add a brief summary of each contribution
# Remember to sign comments on talk pages
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
# Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed)
# User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]
# Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement
# Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand)
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
# Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings)
# Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings)
&lt;!-- 3-level --&gt;
# Overly complicated [[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying
&lt;!-- Conceptual --&gt;
# Messy heading structure – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# The headings lack sufficient incision into, and exposition of, the topic
# Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts.
&lt;!-- Other ---&gt;
# Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections
# The Overview and Conclusion should not use sub-headings
# Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.)
# Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings
# &quot;Introduction&quot; heading isn't necessary – provide this information in Overview or move into subsequent sections
# Cover definition(s) in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information
# Case study doesn't need a separate heading; instead embed case study within relevant sections
# Quiz doesn't need a separate heading; instead embed quiz questions within relevant sections
# Check grammar (e.g,. missing question mark)
# Remove [[wikt:acronym#Noun|acronym]]s from headings
# Remove citations from headings
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure
# Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement
# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
# Reasonably good alignment between focus questions and heading structure, but aim for closer alignment
# Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve.
# Develop closer alignment between sub-title, focus questions, and top-level headings
# Insufficient alignment between sub-title, focus questions, and top-level headings
&lt;!-- GenAI ---&gt;
# Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
# Very good
# Good
# Basic
# Insufficient
# Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- GenAI ---&gt;
# Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
# A scenario or case study is presented in a feature box at the start of this section; I moved an image into the feature box to help attract reader interest
# A scenario or case study is presented in a feature box at the start of this section; add an image to the scenario to help attract reader interest
# Move the scenario or case study into a feature box (with an image) to the start of this section to help engage reader interest
# A scenario or case study is planned
# Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
# A promising description of the problem/topic is planned or presented
# A basic description of the problem/topic is planned or presented
# Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections.
# Add a brief, evocative description of the problem/topic
&lt;!-- Style --&gt;
# Use present, rather than future, tense
# Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios)
&lt;!-- Focus questions --&gt;
# Focus questions are aligned with sub-title and top-level headings
# Reasonably good alignment between focus questions and heading structure, but consider closer alignment
# Develop closer alignment between the sub-title, focus questions, and top-level headings
# Use open- rather then close-ended focus questions
# Use single- rather than double-barrelled focus questions
# Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Present focus questions in a feature box at the end of this section
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Excellent – key points are well developed for each section
# Solid development
# Promising development
# Highlight the most relevant theories and synthesise the best research on the topic
# Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content
# Basic development
# Partial development
# Insufficient development
# No development
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
# The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title
# It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title
# It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
# All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
&lt;!-- Writing style --&gt;
# The writing style is clear and easy to follow
# The writing style is generally clear but could be simplified or made more concise
# The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.)
&lt;!-- Theory and research --&gt;
# Good balance of theory and research
# Promising balance of theory and research
# Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research
# Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. 
# Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. 
# Select the best theories about this topic
# Select the best research about this topic
&lt;!-- Citations --&gt;
# Excellent use of citations
# Very good use of citations
# Good use of citations
# Promising use of citations
# Basic use of citations
# Insufficient use of citations
# Non-peer-reviewed sources should be moved to the &quot;External links&quot; section
# Tip: Rather than starting with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence
&lt;!-- Citation style --&gt;
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials)
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year)
# [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min)
&lt;!-- Other --&gt;
# For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings
# ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title.
# Direct quotes need page numbers (APA style) – even better, express the idea in your own words
# Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a &quot;down&quot; style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info]
# Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective
# Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour)
# Move references into the References section. Keep citations in the main body.
# Consider using the [https://unicanberra.instructure.com/courses/15707/external_tools/262?display=borderless Studiosity] service and/or a service like [https://www.grammarly.com/ Grammarly] to help improve the quality of written expression and to check grammatical and spelling errors
&lt;!-- GenAI ---&gt;
# Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]].
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is well developed
# Conclusion is well underway
# Conclusion is underway
# Conclusion is underdeveloped
# Conclusion hasn't been developed
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Excellent - Relevant figure(s) presented, captioned, and cited
# Relevant figure(s) are presented and captioned
# Relevant figure(s) are presented
# The relevance of the figure to the topic is unclear
# A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Caption --&gt;
# Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text
# Figure caption(s) provide(s) a reasonably clear description that is connected with the main text
# Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved
# Figure caption(s) could better explain how the image connects to key points being made in the main text
# Figure caption(s) should include '''Figure X'''. ...
&lt;!-- Cite --&gt;
# Figure(s) are cited at least once in the main text
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
&lt;!-- Size --&gt;
# Consider increasing image size(s) (especially if they have text) to make them easier to view
# Consider decreasing image size(s) to make them less dominant
&lt;!-- Creation --&gt;
# Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Excellent use of scenarios/examples/case studies
# Promising use of scenarios/examples/case studies
# Keep scenarios brief 
# Basic use of scenario/example/case study
# Placeholder use of scenarios/examples/case studies
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
# Excellent use of quiz question(s)
# Promising use of quiz question(s)
# Placeholder use of quiz question(s)
# Place quiz each question in the most relevant section
# Focus the quiz question(s) on the take-home messages
# Consider including quiz question(s) about the take-home messages
&lt;!-- Tables --&gt;
# Excellent use of table(s)
# Promising use of table(s)
# Use APA style for table captions
# Add table caption
# Cite each table at least once in the text
# Include citations for sources of information presented in the table
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Excellent
# Very good
# Good
# Basic
# Insufficient
# To be developed
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
# At least one relevant systematic review and/or meta-analysis has been identified
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- Move --&gt;
# Move Wikipedia links to the &quot;See also&quot; section
# Move non-academic / non-peer reviewed sources to the &quot;External links&quot; section
&lt;!-- Citations --&gt;
# All references need in-text citation
# All citations need to be in the References
# Only include references which have been accessed and read
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## alphabetical order
## capitalisation
## [[Help:Wikitext quick reference|italicisation]]
## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting]
## make doi hyperlinks active (i.e., clickable)
## use dois where available instead of other links
## include hyperlinked dois
## page numbers should be separated by an en-dash (–) rather than a hyphen (-)
# A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic
# Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used
# Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity.
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
## Very good
## Good
## Basic
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## One of two link types provided
### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]]
### Also link to relevant [[w:|Wikipedia]] pages
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters)
## Use alphabetical order
&lt;!-- External links --&gt;
# External links
## Excellent
## Very good
## Good
## Basic
## One of two required external links provided
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Move Wikipedia link(s) to the &quot;See also&quot; section
## Move academic sources into the &quot;References&quot; sections and provide in-text citation
## Only include links directly related to the sub-title
## Target an international audience; Australians only represent 0.33% of the world population
## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link
## Use alphabetical order
## Link to the most relevant external resources about this topic
|9=
&lt;!-- User page --&gt;
# Excellent
# Used effectively
# Very good
# Good
# Basic
# Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
&lt;!-- Description about self --&gt;
# Excellent description about self provided
# Description about self provided
# Brief description about self – consider expanding
# Very brief description about self – consider expanding
# Add description about self
&lt;!-- Links to profile(s) --&gt;
# Link(s) provided to professional profile(s)
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
# Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Add link to book chapter
|10=
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
# Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making:
# One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
#* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]]
#* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum
# None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]].
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
# Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity.
# Well done on creating and uploading your own image!
# Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Descriptions of contributions could be more precise/accurate/detailed
# Add a brief summary of each contribution
# Remember to sign comments on talk pages
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:58, 26 August 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
* [[Template:MEBF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Topic]]
&lt;/noinclude&gt;</text>
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      <text bytes="47551" sha1="k7b9psortldcl1hmup7tqgmm1dyjqgd" xml:space="preserve">&lt;noinclude&gt;{{note|Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise for [[motivation and emotion]].&lt;br&gt;&lt;br&gt;[[Help:Transclusion|Transclude]] on a chapter [[Help:Talk page|talk page]].}}
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==Topic development feedback==
{{RoundBoxTop|theme=8}}
The [[Motivation and emotion/Assessment/Topic|topic development]] has been reviewed according to the [[Motivation and emotion/Assessment/Topic#Marking criteria|marking criteria]]. Written feedback is provided below, plus see the [[Motivation and emotion/Assessment/Topic/Feedback|general feedback]] page. Also check the [[Special:History/{{PAGENAME}}|page history]] for changes made whilst reviewing the plan. If you don't understand the feedback or would like further information, [[Motivation and emotion/Staff|get in touch]] to discuss. Marks are available via {{Motivation and emotion/Canvas}}. Marks are based on the latest version before the due date.
{{RoundBoxBottom}}
{{RoundBoxTop|theme=9}}
[[File:Autoroute icone.svg|right|85px]]
===1. [[Motivation and emotion/Assessment/Topic#Title|Title]]===
{{{1|No comment}}}
===2. [[Motivation and emotion/Assessment/Topic#Headings|Headings]]===
{{{2|No comment}}}
===3. [[Motivation and emotion/Assessment/Topic#Headings|Overview]]===
{{{3|No comment}}}
===4. [[Motivation and emotion/Assessment/Topic#Key points|Key points]]===
{{{4|No comment}}}
===5. [[Motivation and emotion/Assessment/Topic#Figure|Figure]]===
{{{5|No comment}}}
===6. [[Motivation and emotion/Assessment/Topic#Learning feature|Learning feature]]===
{{{6|No comment}}}
===7. [[Motivation and emotion/Assessment/Topic#References|References]]===
{{{7|No comment}}}
===8. [[Motivation and emotion/Assessment/Topic#Resources|Resources]]===
{{{8|No comment}}}
===9. [[Motivation and emotion/Assessment/Topic#User page|User page]]===
{{{9|No comment}}}
===10. [[Motivation and emotion/Assessment/Topic#Social contribution|Social contribution]]===
{{{10|No comment}}}
{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
#
|2=
&lt;!-- Headings --&gt;
#
|3=
&lt;!-- Overview --&gt;
#
|4=
&lt;!-- Key points--&gt;
#
|5=
&lt;!-- Figure --&gt;
#
|6=
&lt;!-- Learning feature --&gt;
#
|7=
&lt;!-- References --&gt;
#
|8=
&lt;!-- Resources --&gt;
#
|9=
&lt;!-- User page --&gt;
#
|10=
&lt;!-- Social contribution --&gt;
#
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
#
|2=
&lt;!-- Headings --&gt;
#|3=
&lt;!-- Overview --&gt;
#
|4=
&lt;!-- Key points--&gt;
#
|5=
&lt;!-- Figure --&gt;
#
|6=
&lt;!-- Learning feature --&gt;
#
|7=
&lt;!-- References --&gt;
#
|8=
&lt;!-- Resources --&gt;
#
|9=
&lt;!-- User page --&gt;
#
|10=
&lt;!-- Social contribution --&gt;
#
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:42, 17 August 2025 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template which includes commonly used feedback comments:
&lt;pre&gt;
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
# Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed)
# User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]
# Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement
# Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand)
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
# Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings)
# Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings)
&lt;!-- 3-level --&gt;
# Overly complicated[[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying
&lt;!-- Conceptual --&gt;
# Messy heading structure – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# The headings lack sufficient incision into, and exposition of, the topic
# Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts.
&lt;!-- Other ---&gt;
# Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections
# The Overview and Conclusion should not use sub-headings
# Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.)
# Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings
# &quot;Introduction&quot; heading isn't necessary – provide this information in Overview or move into subsequent sections
# Cover definition(s) in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information
# Case study doesn't need a separate heading; instead embed case study within relevant sections
# Quiz doesn't need a separate heading; instead embed quiz questions within relevant sections
# Check grammar (e.g,. missing question mark)
# Remove [[wikt:acronym#Noun|acronym]]s from headings
# Remove citations from headings
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure
# Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement
# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
# Reasonably good alignment between focus questions and heading structure, but aim for closer alignment
# Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve.
# Develop closer alignment between sub-title, focus questions, and top-level headings
# Insufficient alignment between sub-title, focus questions, and top-level headings
&lt;!-- GenAI ---&gt;
# Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
# Very good
# Good
# Basic
# Insufficient
# Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- GenAI ---&gt;
# Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
# A scenario or case study is presented in a feature box at the start of this section; I moved an image into the feature box to help attract reader interest
# A scenario or case study is presented in a feature box at the start of this section; add an image to the scenario to help attract reader interest
# Move the scenario or case study into a feature box (with an image) to the start of this section to help engage reader interest
# A scenario or case study is planned
# Make the relevance of the scenario to the topic more clear
# Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
# A promising description of the problem/topic is planned or presented
# A basic description of the problem/topic is planned or presented
# Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections.
# Add a brief, evocative description of the problem/topic
&lt;!-- Style --&gt;
# Use present, rather than future, tense
# Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios)
&lt;!-- Focus questions --&gt;
# Focus questions are aligned with sub-title and top-level headings
# Reasonably good alignment between focus questions and heading structure, but consider closer alignment
# Develop closer alignment between the sub-title, focus questions, and top-level headings
# Use open- rather then close-ended focus questions
# Use single- rather than double-barrelled focus questions
# Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Present focus questions in a feature box at the end of this section
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Excellent – key points are well developed for each section
# Solid development
# Promising development
# Highlight the most relevant theories and synthesise the best research on the topic
# Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content
# Basic development
# Partial development
# Insufficient development
# No development
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
# The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title
# It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title
# It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
# All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
&lt;!-- Writing style --&gt;
# The writing style is clear and easy to follow
# The writing style is generally clear but could be simplified or made more concise
# The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.)
&lt;!-- Theory and research --&gt;
# Good balance of theory and research
# Promising balance of theory and research
# Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research
# Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples.
# Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples.
# Select the best theories about this topic
# Select the best research about this topic
&lt;!-- Citations --&gt;
# Excellent use of citations
# Very good use of citations
# Good use of citations
# Promising use of citations
# Basic use of citations
# Insufficient use of citations
# Non-peer-reviewed sources should be moved to the &quot;External links&quot; section
# Tip: Rather than starting with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence
&lt;!-- Citation style --&gt;
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials)
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year)
# [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min)
&lt;!-- Other --&gt;
# For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings
# ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title.
# Direct quotes need page numbers (APA style) – even better, express the idea in your own words
# Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a &quot;down&quot; style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info]
# Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective
# Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour)
# Move references into the References section. Keep citations in the main body.
# Consider using the [https://unicanberra.instructure.com/courses/15707/external_tools/262?display=borderless Studiosity] service and/or a service like [https://www.grammarly.com/ Grammarly] to help improve the quality of written expression and to check grammatical and spelling errors
&lt;!-- GenAI ---&gt;
# Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]].
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is well developed
# Conclusion is well underway
# Conclusion is underway
# Conclusion is underdeveloped
# Conclusion hasn't been developed
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Excellent - Relevant figure(s) presented, captioned, and cited
# Relevant figure(s) are presented and captioned
# Relevant figure(s) are presented
# The relevance of the figure to the topic is unclear
# A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Caption --&gt;
# Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text
# Figure caption(s) provide(s) a reasonably clear description that is connected with the main text
# Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved
# Figure caption(s) could better explain how the image connects to key points being made in the main text
# Figure caption(s) should include '''Figure X'''. ...
&lt;!-- Cite --&gt;
# Figure(s) are cited at least once in the main text
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
&lt;!-- Size --&gt;
# Consider increasing image size(s) (especially if they have text) to make them easier to view
# Consider decreasing image size(s) to make them less dominant
&lt;!-- Creation --&gt;
# Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Excellent use of scenarios/examples/case studies
# Promising use of scenarios/examples/case studies
# Keep scenarios brief 
# Basic use of scenario/example/case study
# Placeholder use of scenarios/examples/case studies
# Make the relevance of the scenario to the topic more clear
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
# Excellent use of quiz question(s)
# Promising use of quiz question(s)
# Placeholder use of quiz question(s)
# Place quiz each question in the most relevant section
# Focus the quiz question(s) on the take-home messages
# Consider including quiz question(s) about the take-home messages
&lt;!-- Tables --&gt;
# Excellent use of table(s)
# Promising use of table(s)
# Use APA style for table captions
# Add table caption
# Cite each table at least once in the text
# Include citations for sources of information presented in the table
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Excellent
# Very good
# Good
# Basic
# Insufficient
# To be developed
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
# At least one relevant systematic review and/or meta-analysis has been identified
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- Move --&gt;
# Move Wikipedia links to the &quot;See also&quot; section
# Move non-academic / non-peer reviewed sources to the &quot;External links&quot; section
&lt;!-- Citations --&gt;
# All references need in-text citation
# All citations need to be in the References
# Only include references which have been accessed and read
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## alphabetical order
## capitalisation
## [[Help:Wikitext quick reference|italicisation]]
## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting]
## make doi hyperlinks active (i.e., clickable)
## use dois where available instead of other links
## include hyperlinked dois
## page numbers should be separated by an en-dash (–) rather than a hyphen (-)
# A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic
# Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used
# Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity.
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
## Very good
## Good
## Basic
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## One of two link types provided
### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]]
### Also link to relevant [[w:|Wikipedia]] pages
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters)
## Use alphabetical order
&lt;!-- External links --&gt;
# External links
## Excellent
## Very good
## Good
## Basic
## One of two required external links provided
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Move Wikipedia link(s) to the &quot;See also&quot; section
## Move academic sources into the &quot;References&quot; sections and provide in-text citation
## Only include links directly related to the sub-title
## Target an international audience; Australians only represent 0.33% of the world population
## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link
## Use alphabetical order
## Link to the most relevant external resources about this topic
|9=
&lt;!-- User page --&gt;
# Excellent
# Used effectively
# Very good
# Good
# Basic but effective
# Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
&lt;!-- Description about self --&gt;
# Excellent description about self provided
# Description about self provided
# Brief description about self – consider expanding
# Very brief description about self – consider expanding
# Add description about self
&lt;!-- Links to profile(s) --&gt;
# Link(s) provided to professional profile(s)
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
# Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Add link to book chapter
|10=
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
# Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making:
# One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
#* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]]
#* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum
# None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]].
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
# Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity.
# Well done on creating and uploading your own image!
# Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Descriptions of contributions could be more precise/accurate/detailed
# Add a brief summary of each contribution
# Remember to sign comments on talk pages
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
# Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed)
# User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]
# Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement
# Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand)
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
# Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings)
# Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings)
&lt;!-- 3-level --&gt;
# Overly complicated [[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying
&lt;!-- Conceptual --&gt;
# Messy heading structure – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# The headings lack sufficient incision into, and exposition of, the topic
# Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts.
&lt;!-- Other ---&gt;
# Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections
# The Overview and Conclusion should not use sub-headings
# Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.)
# Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings
# &quot;Introduction&quot; heading isn't necessary – provide this information in Overview or move into subsequent sections
# Cover definition(s) in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information
# Case study doesn't need a separate heading; instead embed case study within relevant sections
# Quiz doesn't need a separate heading; instead embed quiz questions within relevant sections
# Check grammar (e.g,. missing question mark)
# Remove [[wikt:acronym#Noun|acronym]]s from headings
# Remove citations from headings
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure
# Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement
# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
# Reasonably good alignment between focus questions and heading structure, but aim for closer alignment
# Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve.
# Develop closer alignment between sub-title, focus questions, and top-level headings
# Insufficient alignment between sub-title, focus questions, and top-level headings
&lt;!-- GenAI ---&gt;
# Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
# Very good
# Good
# Basic
# Insufficient
# Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- GenAI ---&gt;
# Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
# A scenario or case study is presented in a feature box at the start of this section; I moved an image into the feature box to help attract reader interest
# A scenario or case study is presented in a feature box at the start of this section; add an image to the scenario to help attract reader interest
# Move the scenario or case study into a feature box (with an image) to the start of this section to help engage reader interest
# A scenario or case study is planned
# Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
# A promising description of the problem/topic is planned or presented
# A basic description of the problem/topic is planned or presented
# Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections.
# Add a brief, evocative description of the problem/topic
&lt;!-- Style --&gt;
# Use present, rather than future, tense
# Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios)
&lt;!-- Focus questions --&gt;
# Focus questions are aligned with sub-title and top-level headings
# Reasonably good alignment between focus questions and heading structure, but consider closer alignment
# Develop closer alignment between the sub-title, focus questions, and top-level headings
# Use open- rather then close-ended focus questions
# Use single- rather than double-barrelled focus questions
# Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Present focus questions in a feature box at the end of this section
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Excellent – key points are well developed for each section
# Solid development
# Promising development
# Highlight the most relevant theories and synthesise the best research on the topic
# Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content
# Basic development
# Partial development
# Insufficient development
# No development
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
# The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title
# It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title
# It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
# All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
&lt;!-- Writing style --&gt;
# The writing style is clear and easy to follow
# The writing style is generally clear but could be simplified or made more concise
# The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.)
&lt;!-- Theory and research --&gt;
# Good balance of theory and research
# Promising balance of theory and research
# Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research
# Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. 
# Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. 
# Select the best theories about this topic
# Select the best research about this topic
&lt;!-- Citations --&gt;
# Excellent use of citations
# Very good use of citations
# Good use of citations
# Promising use of citations
# Basic use of citations
# Insufficient use of citations
# Non-peer-reviewed sources should be moved to the &quot;External links&quot; section
# Tip: Rather than starting with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence
&lt;!-- Citation style --&gt;
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials)
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year)
# [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min)
&lt;!-- Other --&gt;
# For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings
# ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title.
# Direct quotes need page numbers (APA style) – even better, express the idea in your own words
# Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a &quot;down&quot; style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info]
# Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective
# Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour)
# Move references into the References section. Keep citations in the main body.
# Consider using the [https://unicanberra.instructure.com/courses/15707/external_tools/262?display=borderless Studiosity] service and/or a service like [https://www.grammarly.com/ Grammarly] to help improve the quality of written expression and to check grammatical and spelling errors
&lt;!-- GenAI ---&gt;
# Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]].
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is well developed
# Conclusion is well underway
# Conclusion is underway
# Conclusion is underdeveloped
# Conclusion hasn't been developed
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Excellent - Relevant figure(s) presented, captioned, and cited
# Relevant figure(s) are presented and captioned
# Relevant figure(s) are presented
# The relevance of the figure to the topic is unclear
# A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Caption --&gt;
# Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text
# Figure caption(s) provide(s) a reasonably clear description that is connected with the main text
# Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved
# Figure caption(s) could better explain how the image connects to key points being made in the main text
# Figure caption(s) should include '''Figure X'''. ...
&lt;!-- Cite --&gt;
# Figure(s) are cited at least once in the main text
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
&lt;!-- Size --&gt;
# Consider increasing image size(s) (especially if they have text) to make them easier to view
# Consider decreasing image size(s) to make them less dominant
&lt;!-- Creation --&gt;
# Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Excellent use of scenarios/examples/case studies
# Promising use of scenarios/examples/case studies
# Keep scenarios brief 
# Basic use of scenario/example/case study
# Placeholder use of scenarios/examples/case studies
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
# Excellent use of quiz question(s)
# Promising use of quiz question(s)
# Placeholder use of quiz question(s)
# Place quiz each question in the most relevant section
# Focus the quiz question(s) on the take-home messages
# Consider including quiz question(s) about the take-home messages
&lt;!-- Tables --&gt;
# Excellent use of table(s)
# Promising use of table(s)
# Use APA style for table captions
# Add table caption
# Cite each table at least once in the text
# Include citations for sources of information presented in the table
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Excellent
# Very good
# Good
# Basic
# Insufficient
# To be developed
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
# At least one relevant systematic review and/or meta-analysis has been identified
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- Move --&gt;
# Move Wikipedia links to the &quot;See also&quot; section
# Move non-academic / non-peer reviewed sources to the &quot;External links&quot; section
&lt;!-- Citations --&gt;
# All references need in-text citation
# All citations need to be in the References
# Only include references which have been accessed and read
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## alphabetical order
## capitalisation
## [[Help:Wikitext quick reference|italicisation]]
## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting]
## make doi hyperlinks active (i.e., clickable)
## use dois where available instead of other links
## include hyperlinked dois
## page numbers should be separated by an en-dash (–) rather than a hyphen (-)
# A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic
# Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used
# Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity.
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
## Very good
## Good
## Basic
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## One of two link types provided
### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]]
### Also link to relevant [[w:|Wikipedia]] pages
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters)
## Use alphabetical order
&lt;!-- External links --&gt;
# External links
## Excellent
## Very good
## Good
## Basic
## One of two required external links provided
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Move Wikipedia link(s) to the &quot;See also&quot; section
## Move academic sources into the &quot;References&quot; sections and provide in-text citation
## Only include links directly related to the sub-title
## Target an international audience; Australians only represent 0.33% of the world population
## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link
## Use alphabetical order
## Link to the most relevant external resources about this topic
|9=
&lt;!-- User page --&gt;
# Excellent
# Used effectively
# Very good
# Good
# Basic but effective
# Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
&lt;!-- Description about self --&gt;
# Excellent description about self provided
# Description about self provided
# Brief description about self – consider expanding
# Very brief description about self – consider expanding
# Add description about self
&lt;!-- Links to profile(s) --&gt;
# Link(s) provided to professional profile(s)
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
# Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Add link to book chapter
|10=
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
# Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making:
# One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
#* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]]
#* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum
# None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]].
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
# Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity.
# Well done on creating and uploading your own image!
# Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Descriptions of contributions could be more precise/accurate/detailed
# Add a brief summary of each contribution
# Remember to sign comments on talk pages
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:58, 26 August 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
* [[Template:MEBF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Topic]]
&lt;/noinclude&gt;</text>
      <sha1>k7b9psortldcl1hmup7tqgmm1dyjqgd</sha1>
    </revision>
    <revision>
      <id>2829860</id>
      <parentid>2829843</parentid>
      <timestamp>2026-08-31T09:47:22Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment># Balance theoretical content with critical synthesis of relevant research</comment>
      <origin>2829860</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="47701" sha1="iffc8ozoh9u7s05y4zgyn0kf8753e07" xml:space="preserve">&lt;noinclude&gt;{{note|Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise for [[motivation and emotion]].&lt;br&gt;&lt;br&gt;[[Help:Transclusion|Transclude]] on a chapter [[Help:Talk page|talk page]].}}
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==Topic development feedback==
{{RoundBoxTop|theme=8}}
The [[Motivation and emotion/Assessment/Topic|topic development]] has been reviewed according to the [[Motivation and emotion/Assessment/Topic#Marking criteria|marking criteria]]. Written feedback is provided below, plus see the [[Motivation and emotion/Assessment/Topic/Feedback|general feedback]] page. Also check the [[Special:History/{{PAGENAME}}|page history]] for changes made whilst reviewing the plan. If you don't understand the feedback or would like further information, [[Motivation and emotion/Staff|get in touch]] to discuss. Marks are available via {{Motivation and emotion/Canvas}}. Marks are based on the latest version before the due date.
{{RoundBoxBottom}}
{{RoundBoxTop|theme=9}}
[[File:Autoroute icone.svg|right|85px]]
===1. [[Motivation and emotion/Assessment/Topic#Title|Title]]===
{{{1|No comment}}}
===2. [[Motivation and emotion/Assessment/Topic#Headings|Headings]]===
{{{2|No comment}}}
===3. [[Motivation and emotion/Assessment/Topic#Headings|Overview]]===
{{{3|No comment}}}
===4. [[Motivation and emotion/Assessment/Topic#Key points|Key points]]===
{{{4|No comment}}}
===5. [[Motivation and emotion/Assessment/Topic#Figure|Figure]]===
{{{5|No comment}}}
===6. [[Motivation and emotion/Assessment/Topic#Learning feature|Learning feature]]===
{{{6|No comment}}}
===7. [[Motivation and emotion/Assessment/Topic#References|References]]===
{{{7|No comment}}}
===8. [[Motivation and emotion/Assessment/Topic#Resources|Resources]]===
{{{8|No comment}}}
===9. [[Motivation and emotion/Assessment/Topic#User page|User page]]===
{{{9|No comment}}}
===10. [[Motivation and emotion/Assessment/Topic#Social contribution|Social contribution]]===
{{{10|No comment}}}
{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
#
|2=
&lt;!-- Headings --&gt;
#
|3=
&lt;!-- Overview --&gt;
#
|4=
&lt;!-- Key points--&gt;
#
|5=
&lt;!-- Figure --&gt;
#
|6=
&lt;!-- Learning feature --&gt;
#
|7=
&lt;!-- References --&gt;
#
|8=
&lt;!-- Resources --&gt;
#
|9=
&lt;!-- User page --&gt;
#
|10=
&lt;!-- Social contribution --&gt;
#
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
#
|2=
&lt;!-- Headings --&gt;
#|3=
&lt;!-- Overview --&gt;
#
|4=
&lt;!-- Key points--&gt;
#
|5=
&lt;!-- Figure --&gt;
#
|6=
&lt;!-- Learning feature --&gt;
#
|7=
&lt;!-- References --&gt;
#
|8=
&lt;!-- Resources --&gt;
#
|9=
&lt;!-- User page --&gt;
#
|10=
&lt;!-- Social contribution --&gt;
#
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:42, 17 August 2025 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template which includes commonly used feedback comments:
&lt;pre&gt;
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
# Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed)
# User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]
# Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement
# Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand)
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
# Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings)
# Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings)
&lt;!-- 3-level --&gt;
# Overly complicated[[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying
&lt;!-- Conceptual --&gt;
# Messy heading structure – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# The headings lack sufficient incision into, and exposition of, the topic
# Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts.
&lt;!-- Other ---&gt;
# Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections
# The Overview and Conclusion should not use sub-headings
# Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.)
# Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings
# &quot;Introduction&quot; heading isn't necessary – provide this information in Overview or move into subsequent sections
# Cover definition(s) in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information
# Case study doesn't need a separate heading; instead embed case study within relevant sections
# Quiz doesn't need a separate heading; instead embed quiz questions within relevant sections
# Check grammar (e.g,. missing question mark)
# Remove [[wikt:acronym#Noun|acronym]]s from headings
# Remove citations from headings
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure
# Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement
# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
# Reasonably good alignment between focus questions and heading structure, but aim for closer alignment
# Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve.
# Develop closer alignment between sub-title, focus questions, and top-level headings
# Insufficient alignment between sub-title, focus questions, and top-level headings
&lt;!-- GenAI ---&gt;
# Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
# Very good
# Good
# Basic
# Insufficient
# Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- GenAI ---&gt;
# Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
# A scenario or case study is presented in a feature box at the start of this section; I moved an image into the feature box to help attract reader interest
# A scenario or case study is presented in a feature box at the start of this section; add an image to the scenario to help attract reader interest
# Move the scenario or case study into a feature box (with an image) to the start of this section to help engage reader interest
# A scenario or case study is planned
# Make the relevance of the scenario to the topic more clear
# Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
# A promising description of the problem/topic is planned or presented
# A basic description of the problem/topic is planned or presented
# Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections.
# Add a brief, evocative description of the problem/topic
&lt;!-- Style --&gt;
# Use present, rather than future, tense
# Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios)
&lt;!-- Focus questions --&gt;
# Focus questions are aligned with sub-title and top-level headings
# Reasonably good alignment between focus questions and heading structure, but consider closer alignment
# Develop closer alignment between the sub-title, focus questions, and top-level headings
# Use open- rather then close-ended focus questions
# Use single- rather than double-barrelled focus questions
# Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Present focus questions in a feature box at the end of this section
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Excellent – key points are well developed for each section
# Solid development
# Promising development
# Highlight the most relevant theories and synthesise the best research on the topic
# Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content
# Basic development
# Partial development
# Insufficient development
# No development
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
# The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title
# It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title
# It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
# All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
&lt;!-- Writing style --&gt;
# The writing style is clear and easy to follow
# The writing style is generally clear but could be simplified or made more concise
# The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.)
&lt;!-- Theory and research --&gt;
# Good balance of theory and research
# Promising balance of theory and research
# Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research
# Balance theoretical content with critical synthesis of relevant research
# Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples.
# Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples.
# Select the best theories about this topic
# Select the best research about this topic
&lt;!-- Citations --&gt;
# Excellent use of citations
# Very good use of citations
# Good use of citations
# Promising use of citations
# Basic use of citations
# Insufficient use of citations
# Non-peer-reviewed sources should be moved to the &quot;External links&quot; section
# Tip: Rather than starting with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence
&lt;!-- Citation style --&gt;
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials)
# Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year)
# [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min)
&lt;!-- Other --&gt;
# For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings
# ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title.
# Direct quotes need page numbers (APA style) – even better, express the idea in your own words
# Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a &quot;down&quot; style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info]
# Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective
# Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour)
# Move references into the References section. Keep citations in the main body.
# Consider using the [https://unicanberra.instructure.com/courses/15707/external_tools/262?display=borderless Studiosity] service and/or a service like [https://www.grammarly.com/ Grammarly] to help improve the quality of written expression and to check grammatical and spelling errors
&lt;!-- GenAI ---&gt;
# Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]].
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is well developed
# Conclusion is well underway
# Conclusion is underway
# Conclusion is underdeveloped
# Conclusion hasn't been developed
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Excellent - Relevant figure(s) presented, captioned, and cited
# Relevant figure(s) are presented and captioned
# Relevant figure(s) are presented
# The relevance of the figure to the topic is unclear
# A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Caption --&gt;
# Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text
# Figure caption(s) provide(s) a reasonably clear description that is connected with the main text
# Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved
# Figure caption(s) could better explain how the image connects to key points being made in the main text
# Figure caption(s) should include '''Figure X'''. ...
&lt;!-- Cite --&gt;
# Figure(s) are cited at least once in the main text
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
&lt;!-- Size --&gt;
# Consider increasing image size(s) (especially if they have text) to make them easier to view
# Consider decreasing image size(s) to make them less dominant
&lt;!-- Creation --&gt;
# Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
# Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Excellent use of scenarios/examples/case studies
# Promising use of scenarios/examples/case studies
# Keep scenarios brief 
# Basic use of scenario/example/case study
# Placeholder use of scenarios/examples/case studies
# Make the relevance of the scenario to the topic more clear
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
# Excellent use of quiz question(s)
# Promising use of quiz question(s)
# Placeholder use of quiz question(s)
# Place quiz each question in the most relevant section
# Focus the quiz question(s) on the take-home messages
# Consider including quiz question(s) about the take-home messages
&lt;!-- Tables --&gt;
# Excellent use of table(s)
# Promising use of table(s)
# Use APA style for table captions
# Add table caption
# Cite each table at least once in the text
# Include citations for sources of information presented in the table
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Excellent
# Very good
# Good
# Basic
# Insufficient
# To be developed
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
# At least one relevant systematic review and/or meta-analysis has been identified
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- Move --&gt;
# Move Wikipedia links to the &quot;See also&quot; section
# Move non-academic / non-peer reviewed sources to the &quot;External links&quot; section
&lt;!-- Citations --&gt;
# All references need in-text citation
# All citations need to be in the References
# Only include references which have been accessed and read
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## alphabetical order
## capitalisation
## [[Help:Wikitext quick reference|italicisation]]
## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting]
## make doi hyperlinks active (i.e., clickable)
## use dois where available instead of other links
## include hyperlinked dois
## page numbers should be separated by an en-dash (–) rather than a hyphen (-)
# A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic
# Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used
# Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity.
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
## Very good
## Good
## Basic
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## One of two link types provided
### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]]
### Also link to relevant [[w:|Wikipedia]] pages
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters)
## Use alphabetical order
&lt;!-- External links --&gt;
# External links
## Excellent
## Very good
## Good
## Basic
## One of two required external links provided
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Move Wikipedia link(s) to the &quot;See also&quot; section
## Move academic sources into the &quot;References&quot; sections and provide in-text citation
## Only include links directly related to the sub-title
## Target an international audience; Australians only represent 0.33% of the world population
## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use [[w:Letter case#Sentence casing|sentence casing]]
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Include source in brackets after link
## Use alphabetical order
## Link to the most relevant external resources about this topic
|9=
&lt;!-- User page --&gt;
# Excellent
# Used effectively
# Very good
# Good
# Basic but effective
# Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
&lt;!-- Description about self --&gt;
# Excellent description about self provided
# Description about self provided
# Brief description about self – consider expanding
# Very brief description about self – consider expanding
# Add description about self
&lt;!-- Links to profile(s) --&gt;
# Link(s) provided to professional profile(s)
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
# Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Add link to book chapter
|10=
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
# Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making:
# One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
#* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]]
#* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum
# None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]].
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
# Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity.
# Well done on creating and uploading your own image!
# Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Descriptions of contributions could be more precise/accurate/detailed
# Add a brief summary of each contribution
# Remember to sign comments on talk pages
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
# Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed)
# User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]
# Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement
# Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand)
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
# Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings)
# Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings)
&lt;!-- 3-level --&gt;
# Overly complicated [[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying
&lt;!-- Conceptual --&gt;
# Messy heading structure – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# The headings lack sufficient incision into, and exposition of, the topic
# Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts.
&lt;!-- Other ---&gt;
# Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections
# The Overview and Conclusion should not use sub-headings
# Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.)
# Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings
# &quot;Introduction&quot; heading isn't necessary – provide this information in Overview or move into subsequent sections
# Cover definition(s) in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information
# Case study doesn't need a separate heading; instead embed case study within relevant sections
# Quiz doesn't need a separate heading; instead embed quiz questions within relevant sections
# Check grammar (e.g,. missing question mark)
# Remove [[wikt:acronym#Noun|acronym]]s from headings
# Remove citations from headings
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure
# Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement
# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
# Reasonably good alignment between focus questions and heading structure, but aim for closer alignment
# Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve.
# Develop closer alignment between sub-title, focus questions, and top-level headings
# Insufficient alignment between sub-title, focus questions, and top-level headings
&lt;!-- GenAI ---&gt;
# Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
# Very good
# Good
# Basic
# Insufficient
# Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- GenAI ---&gt;
# Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
# A scenario or case study is presented in a feature box at the start of this section; I moved an image into the feature box to help attract reader interest
# A scenario or case study is presented in a feature box at the start of this section; add an image to the scenario to help attract reader interest
# Move the scenario or case study into a feature box (with an image) to the start of this section to help engage reader interest
# A scenario or case study is planned
# Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
# A promising description of the problem/topic is planned or presented
# A basic description of the problem/topic is planned or presented
# Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections.
# Add a brief, evocative description of the problem/topic
&lt;!-- Style --&gt;
# Use present, rather than future, tense
# Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios)
&lt;!-- Focus questions --&gt;
# Focus questions are aligned with sub-title and top-level headings
# Reasonably good alignment between focus questions and heading structure, but consider closer alignment
# Develop closer alignment between the sub-title, focus questions, and top-level headings
# Use open- rather then close-ended focus questions
# Use single- rather than double-barrelled focus questions
# Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Present focus questions in a feature box at the end of this section
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Excellent – key points are well developed for each section
# Solid development
# Promising development
# Highlight the most relevant theories and synthesise the best research on the topic
# Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content
# Basic development
# Partial development
# Insufficient development
# No development
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
# The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title
# It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title
# It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
# All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
&lt;!-- Writing style --&gt;
# The writing style is clear and easy to follow
# The writing style is generally clear but could be simplified or made more concise
# The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.)
&lt;!-- Theory and research --&gt;
# Good balance of theory and research
# Promising balance of theory and research
# Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research
# Balance theoretical content with critical synthesis of relevant research
# Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. 
# Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. 
# Select the best theories about this topic
# Select the best research about this topic
&lt;!-- Citations --&gt;
# Excellent use of citations
# Very good use of citations
# Good use of citations
# Promising use of citations
# Basic use of citations
# Insufficient use of citations
# Non-peer-reviewed sources should be moved to the &quot;External links&quot; section
# Tip: Rather than starting with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence
&lt;!-- Citation style --&gt;
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# [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min)
&lt;!-- Other --&gt;
# For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings
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&lt;!-- Link to book chapter --&gt;
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|10=
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#* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum
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# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
# Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity.
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}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:58, 26 August 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
* [[Template:MEBF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Topic]]
&lt;/noinclude&gt;</text>
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    <title>The John Snow Prediabetes Institute</title>
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      <text bytes="5187" sha1="t9nd56fzqdhihiap9b11ic5u2ggpv08" xml:space="preserve">The John Snow prediabetes Institute is an international research network focused on prediabetes remission (prevention) , early risk …identification, and metabolic health education. (https://w.wiki/Skm7).

'''Background:''' 
Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight (General Research Draft). &lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt; All can lower their risks by staying physical active and eating well. Early diagnosis and education of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes. The arteriosclerosis starts in the prediabetes stage. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained. For early identification of the risks we propose to register weight and height (BMI), the fasting blood sugar (glucometer), blood pressure, age, gender in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools followed by giving educational materials.The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer &lt;ref&gt;https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 &lt;/ref&gt; and self-evaluation of diet and physical activity. Educational materials come from the international diabetes organisations e.g the ADA:&lt;ref&gt;https://professional.diabetes.org/diabetes-support-resources&lt;/ref&gt; and the Health Promoting Schools.&lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;[[File:Lifestyle Medicine Pillars.png|250px|right|&lt;big&gt;
Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars&lt;/big&gt; ]]

&lt;big&gt;'''Prevalence studies'''  &lt;/big&gt;

[[/The Maritime Health database design // |1.1 The Maritime Health database design]] 

1.2  Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

1.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

1.4. School childrens` Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

1.5  Excel recoding (pre diabetes, hypertension etc) &lt;ref&gt;https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG&lt;/ref&gt;

&lt;big&gt;&lt;br /&gt;
'''Intervention studies''' English 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

'''Organisation''' 

- [[/The John Snow Institute bylaws /]] 

- [[/|Manila conference Oct. 2026]] 

- General research pr[[Category:Prediabetes ]]otoco[[Category:Prediabetes ]]l draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

- Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;

'''Publications'''
&lt;ref name=&quot;:0&quot;&gt; https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf &lt;/ref&gt;

'''The John Snow Prediabetes Institute Research Network:'''
&lt;small&gt;Cordinator and Director MBA Christian Acheampong, Turkey, Prof. Magda Medir Mb, Spain, Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark&lt;/small&gt;

==References==

&lt;references /&gt;
Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
&lt;references /&gt;</text>
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'''Binary Grammar'''

'''Motivation'''

Universal grammar has dominated linguistics courses in the United States. Many linguistics courses were only created after Noam Chomsky’s seminal work in trying to define what a grammar formally using logic. He critiqued B. F. Skinner’s application of behavioural psychology to linguistics&lt;ref&gt;{{Cite web|url=https://chomsky.info/1967____/|title=Review of B. F. Skinner’s Verbal Behavior|website=chomsky.info|access-date=2026-08-30}}&lt;/ref&gt;.

The distinction between “E-language” and “I-language” has created a useful language to talk about language itself. That way, linguistics could seen as the study of the innate faculties of the human mind regarding I-language, which Chomsky defined as a proper object of scientific study&lt;ref&gt;{{Cite web|url=https://www.rep.routledge.com/articles/biographical/chomsky-noam-1928/v-1/sections/i-language-versus-e-language|title=Chomsky, Noam (1928–) - Routledge Encyclopedia of Philosophy|website=www.rep.routledge.com|language=en|access-date=2026-08-30}}&lt;/ref&gt;.

This issue is complicated by the fact that universal grammar has a vague definition and there is plenty of contradicting evidence against some parts of it&lt;ref&gt;{{Cite journal|last=Dąbrowska|first=Ewa|date=2015|title=What exactly is Universal Grammar, and has anyone seen it?|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4477053/|journal=Frontiers in Psychology|volume=6|pages=852|doi=10.3389/fpsyg.2015.00852|issn=1664-1078|pmc=4477053|pmid=26157406}}&lt;/ref&gt;.

This has made universal grammar a controversial topic among members of the scientific community.

This is not just supposed to be a critique of universal grammar, because universal grammar has already very criticized by many scientists.

This is presented as a research programme in the sense of Imre Lakatos&lt;ref name=&quot;:0&quot;&gt;{{Cite book|url=https://plato.stanford.edu/archives/sum2026/entries/lakatos/|title=Imre Lakatos|last=Musgrave|first=Alan|last2=Pigden|first2=Charles|date=2026|publisher=Metaphysics Research Lab, Stanford University|editor-last=Zalta|editor-first=Edward N.|edition=Summer 2026|editor-last2=Nodelman|editor-first2=Uri}}&lt;/ref&gt;.

The protective belt could be about, in this case, would be how would language be defined using this research programme. The positive heuristic is to define a research programme that does not have to posit merge or tree structures associated with traditional linguistics

The negative heuristic is not to introduce ideas unnecessary to explain human language or, in other words, to minimize the number of assumptions one has to make ideally.

'''Definition'''

In this sense, binary is used to mean the two dualism between “0” and “1” or “true” or “false”.

A neuron, in this theory, would not be a cell of the brain, but would either carry 0 or 1, with no ambiguity. It is not a claim of biology, but merely a mathematical model.

A grammar includes both a vocabulary and an iterative function, which could be modelled as a mathematical or computer function and a vocabulary function which matches the state of the neurons with the symbol.

The iterative function takes randomness and the previous state of the program or the initial state of the program. Depending on the output of the function, the program halts or gets to a next state.

The vocabulary function takes a state of neurons and returns a vocabulary item.

There are “symbols” which map to the smallest possible unit in binary grammar. There is no need to posit a difference between “morphology” and “syntax” in this frame work.

Example of a binary grammar coded in Python 3:&lt;syntaxhighlight lang=&quot;python3&quot;&gt;
#!/usr/bin/env python3

import random

a = True

b = math.randint(0,1)

c = False

d = False

e = False

while True:

   if a:

       if b:

           print( &quot;A&quot; )

       else:

           print( &quot;B&quot; )    

   if c:

       print( &quot;is&quot; )

   
   if d:

       print( &quot;not&quot; )

   
   if c:

       c = False

       a = True

   elif d:

       d = False

       c = True

   elif a:

       if e:

           break

       a = False

       e = True

       b = not b

       if (random.randint(0,1)):

           c = True

           d = False

       else:

           d = True

           c = False

Therefore:

A not is B.

A is B.

B not is A.

B is A.

are all valid constructions under this simple grammar example.
&lt;/syntaxhighlight&gt;'''Formalizations'''

It is a program such that it takes as input Sx and R wherein R means &quot;random&quot; and it either halts or it actually presents an output.

'''Objections'''

# '''Does this means there is a only a final number of states?''' That is a central point of this programme. It is to reject infinite complexity in the name of the biologically possible and to anchor grammar in a finite state machine.
# '''Does it make falsifiable claims in the sense of Karl Popper?''' First, strict falsifiability has already been questioned by later philosophers like Thomas Kuhn&lt;ref&gt;{{Cite book|url=https://plato.stanford.edu/archives/fall2025/entrieshomas-kuhn/|title=Thomas Kuhn|last=Bird|first=Alexander|date=2025|publisher=Metaphysics Research Lab, Stanford University|editor-last=Zalta|editor-first=Edward N.|edition=Fall 2025|editor-last2=Nodelman|editor-first2=Uri}}&lt;/ref&gt; and Imre Lakatos&lt;ref name=&quot;:0&quot; /&gt;. Scientists note he did not understand psychoanalysis as a psychotherapy.&lt;ref&gt;{{Cite journal|last=Grant|first=Don C.|last2=Harari|first2=Edwin|date=2005-06|title=Psychoanalysis, science and the seductive theory of Karl Popper|url=https://pubmed.ncbi.nlm.nih.gov/15943645|journal=The Australian and New Zealand Journal of Psychiatry|volume=39|issue=6|pages=446–452|doi=10.1080/j.1440-1614.2005.01602.x|issn=0004-8674|pmid=15943645}}&lt;/ref&gt; Second, the central claim is that this is an effective way of thinking about human grammar more so than the traditional tree model and also one of directionality: Subject -&gt; Verb -&gt; Object rather than Sentence { Subject Predicate { Verb/Object.
# '''How to deal with recursion?''' A binary grammar would be able to create infinite sentences, depending on the exact contraints. For instance, it is possible to define a grammar such that there are nouns and propositions. It would be possible to define that a sentence starts without a noun and then what follows would be prepositions followed by nouns until it halts, so it is possible to define recursion.</text>
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'''Gramática binária'''

'''Motivação'''

A gramática universal têm dominado cursos de linguística nos Estados Unidos. Muitos deles só foram criados após o trabalho seminal de Noam Chomsky em tentar definir o que é uma gramática usando a lógica. Ele criticou o trabalho de B. F. Skinner no qual aplica o behaviorismo à linguística&lt;ref&gt;{{Cite web|url=https://chomsky.info/1967____/|title=Review of B. F. Skinner’s Verbal Behavior|website=chomsky.info|access-date=2026-08-30}}&lt;/ref&gt;.

A distinção entre língua-E e língua-I criou uma linguagem util para falar da própria da linguagem. Desse jeito, a linguistica poderia ser vista como o estudo das faculdades inatas do cérebro humano em relação a língua-I, que Chomsky definiu como um objeto próprio de estudo científico&lt;ref&gt;{{Cite web|url=https://www.rep.routledge.com/articles/biographical/chomsky-noam-1928/v-1/sections/i-language-versus-e-language|title=Chomsky, Noam (1928–) - Routledge Encyclopedia of Philosophy|website=www.rep.routledge.com|language=en|access-date=2026-08-30}}&lt;/ref&gt;.

Isso é complicado pelo fato que a gramática universal tem uma definição vaga e há muita evidência contra formas específicas dela&lt;ref&gt;{{Cite journal|last=Dąbrowska|first=Ewa|date=2015|title=What exactly is Universal Grammar, and has anyone seen it?|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4477053/|journal=Frontiers in Psychology|volume=6|pages=852|doi=10.3389/fpsyg.2015.00852|issn=1664-1078|pmc=4477053|pmid=26157406}}&lt;/ref&gt;.

Isso fez a gramática universal algo controverso dentro da comunidade científica.

Isso não é apenas uma crítica da gramática universal, pois ela foi criticada por vários cientistas já.

O modelo é apresentado como um programa de pesquisa no sentido do filósofo húngaro Imre Lakatos&lt;ref name=&quot;:0&quot;&gt;{{Cite book|url=https://plato.stanford.edu/archives/sum2026/entries/lakatos/|title=Imre Lakatos|last=Musgrave|first=Alan|last2=Pigden|first2=Charles|date=2026|publisher=Metaphysics Research Lab, Stanford University|editor-last=Zalta|editor-first=Edward N.|edition=Summer 2026|editor-last2=Nodelman|editor-first2=Uri}}&lt;/ref&gt;, que via programas de pesquisa como degenerativos ou progressivos.

O cinto protetivo poderia ser, nesse caso, sobre as especificidades de como a linguagem seria definido sobre a visão proposta. A heurística positiva é definir um programa de pesquisa que não tem que acreditar em modelos de árvore ou em uma ideia como merge.

A heurística negativa seria não introduzir ideias desnecesárias para a explicar a linguagem humana. Ou seja, a ideia é não presumir o que não é necessário presumir e não introduzir ideias que contradizem com como o cérebro humano age.

'''Definition'''

Nesse sentido, binário é usado para significar o dualismo entre “0” and “1” or “verdadeiro&quot; ou &quot;falso&quot;.

Um neurônio, nessa teoria, não seria uma célula do cérebro, mas carregaria 0 ou 1, sem ambiguidade. Não é uma asserção de biologia, mas uma ideia matemática para a modelagem da linguagem.

A gramática inclui tanto uma função iterativa quanto uma de vocabulário. A função iterativa leva como entrada o primeiro estado ou o estado anterior e para ou cria um novo estado com a consideração da aleatoridade. A função de vocabulário tem como entrada um estado e como saida um símbolo.

Há os “símbolos&quot; que mapeam a menor unidade na gramática binária. Não há necessidade para afirmar uma diferença entre a morfologia e a sintaxe nesse paradigma.

Exemplo de uma gramática binária codificada em Python 3:&lt;syntaxhighlight lang=&quot;python3&quot;&gt;
#!/usr/bin/env python3

import random

a = True

b = math.randint(0,1)

c = False

d = False

e = False

while True:

   if a:

       if b:

           print( &quot;A&quot; )

       else:

           print( &quot;B&quot; )    

   if c:

       print( &quot;is&quot; )

   
   if d:

       print( &quot;not&quot; )

   
   if c:

       c = False

       a = True

   elif d:

       d = False

       c = True

   elif a:

       if e:

           break

       a = False

       e = True

       b = not b

       if (random.randint(0,1)):

           c = True

           d = False

       else:

           d = True

           c = False

#Portanto são construções válidas:

# (a) A not is B.

# (b) A is B.

# (c) B not is A.

# (d) B is A.

&lt;/syntaxhighlight&gt;'''Objeções'''

# '''Isso significa que há apenas um número limitado de estados?''' Esse é um ponto central desse programa. A ideia é ancorar a gramática no que é biologicamente possível e numa máquina que diverge das máquinas de estado tradicionais.
# '''Faz asserções falseáveis no sentido do celébro filósofo Karl Popper?''' Primeiramente, a falseabilidade estritia já foi questionada por filósofos posteriores como Thomas Kuhn e Imre Lakatos&lt;ref&gt;{{Cite book|url=https://plato.stanford.edu/archives/fall2025/entrieshomas-kuhn/|title=Thomas Kuhn|last=Bird|first=Alexander|date=2025|publisher=Metaphysics Research Lab, Stanford University|editor-last=Zalta|editor-first=Edward N.|edition=Fall 2025|editor-last2=Nodelman|editor-first2=Uri}}&lt;/ref&gt;&lt;ref name=&quot;:0&quot; /&gt;. Cientistas notam que ele não entendeu a psicanálise como psicoterapia.&lt;ref&gt;{{Cite journal|last=Grant|first=Don C.|last2=Harari|first2=Edwin|date=2005-06|title=Psychoanalysis, science and the seductive theory of Karl Popper|url=https://pubmed.ncbi.nlm.nih.gov/15943645|journal=The Australian and New Zealand Journal of Psychiatry|volume=39|issue=6|pages=446–452|doi=10.1080/j.1440-1614.2005.01602.x|issn=0004-8674|pmid=15943645}}&lt;/ref&gt; Segundamente, a asserção central é que um jeito mais efetivo de considerar a gramática muito mais que o modelo de árvore tradicional e também de um jeito direcional: Sujeito -&gt; Verbo -&gt; Objeto em vez de Sentença { Sujeito Predicado { Verbo/Objeto.
# '''Comop lidar com a recursão?''' A gramática binária poderia criar sentenças infinitas, dependo das condições necessárias. Por exemplo, é possível definir uma gramática de modo que há substantivos e preposições. É possível definir que a sentença começa com um substantivo e o que segue são interpolações de preposições com substantivos, eventualmente culminando num final de frase. Por exemplo, &quot;Pessoa do bem da comunidade de meio...&quot;. Essa seria uma maneira de definir uma gramática que atende a uma forma mais simples de recursão. A recursão não implica necessariamente em orações subordinadas.</text>
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'''Gramática binária: uma ideia alternativa à gramática universal e árvores de sintaxe.'''

'''Motivação'''

A gramática universal têm dominado cursos de linguística nos Estados Unidos. Muitos deles só foram criados após o trabalho seminal de Noam Chomsky em tentar definir o que é uma gramática usando a lógica. Ele criticou o trabalho de B. F. Skinner no qual aplica o behaviorismo à linguística&lt;ref&gt;{{Cite web|url=https://chomsky.info/1967____/|title=Review of B. F. Skinner’s Verbal Behavior|website=chomsky.info|access-date=2026-08-30}}&lt;/ref&gt;.

A distinção entre língua-E e língua-I criou uma linguagem util para falar da própria da linguagem. Desse jeito, a linguistica poderia ser vista como o estudo das faculdades inatas do cérebro humano em relação a língua-I, que Chomsky definiu como um objeto próprio de estudo científico&lt;ref&gt;{{Cite web|url=https://www.rep.routledge.com/articles/biographical/chomsky-noam-1928/v-1/sections/i-language-versus-e-language|title=Chomsky, Noam (1928–) - Routledge Encyclopedia of Philosophy|website=www.rep.routledge.com|language=en|access-date=2026-08-30}}&lt;/ref&gt;.

Isso é complicado pelo fato que a gramática universal tem uma definição vaga e há muita evidência contra formas específicas dela&lt;ref&gt;{{Cite journal|last=Dąbrowska|first=Ewa|date=2015|title=What exactly is Universal Grammar, and has anyone seen it?|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4477053/|journal=Frontiers in Psychology|volume=6|pages=852|doi=10.3389/fpsyg.2015.00852|issn=1664-1078|pmc=4477053|pmid=26157406}}&lt;/ref&gt;.

Isso fez a gramática universal algo controverso dentro da comunidade científica.

Isso não é apenas uma crítica da gramática universal, pois ela foi criticada por vários cientistas já.

O modelo é apresentado como um programa de pesquisa no sentido do filósofo húngaro Imre Lakatos&lt;ref name=&quot;:0&quot;&gt;{{Cite book|url=https://plato.stanford.edu/archives/sum2026/entries/lakatos/|title=Imre Lakatos|last=Musgrave|first=Alan|last2=Pigden|first2=Charles|date=2026|publisher=Metaphysics Research Lab, Stanford University|editor-last=Zalta|editor-first=Edward N.|edition=Summer 2026|editor-last2=Nodelman|editor-first2=Uri}}&lt;/ref&gt;, que via programas de pesquisa como degenerativos ou progressivos.

O cinto protetivo poderia ser, nesse caso, sobre as especificidades de como a linguagem seria definido sobre a visão proposta. A heurística positiva é definir um programa de pesquisa que não tem que acreditar em modelos de árvore ou em uma ideia como merge.

A heurística negativa seria não introduzir ideias desnecesárias para a explicar a linguagem humana. Ou seja, a ideia é não presumir o que não é necessário presumir e não introduzir ideias que contradizem com como o cérebro humano age.

'''Definition'''

Nesse sentido, binário é usado para significar o dualismo entre “0” and “1” or “verdadeiro&quot; ou &quot;falso&quot;.

Um neurônio, nessa teoria, não seria uma célula do cérebro, mas carregaria 0 ou 1, sem ambiguidade. Não é uma asserção de biologia, mas uma ideia matemática para a modelagem da linguagem.

A gramática inclui tanto uma função iterativa quanto uma de vocabulário. A função iterativa leva como entrada o primeiro estado ou o estado anterior e para ou cria um novo estado com a consideração da aleatoridade. A função de vocabulário tem como entrada um estado e como saida um símbolo.

Há os “símbolos&quot; que mapeam a menor unidade na gramática binária. Não há necessidade para afirmar uma diferença entre a morfologia e a sintaxe nesse paradigma.

Exemplo de uma gramática binária codificada em Python 3:&lt;syntaxhighlight lang=&quot;python3&quot;&gt;
#!/usr/bin/env python3

import random

a = True

b = math.randint(0,1)

c = False

d = False

e = False

while True:

   if a:

       if b:

           print( &quot;A&quot; )

       else:

           print( &quot;B&quot; )    

   if c:

       print( &quot;is&quot; )

   
   if d:

       print( &quot;not&quot; )

   
   if c:

       c = False

       a = True

   elif d:

       d = False

       c = True

   elif a:

       if e:

           break

       a = False

       e = True

       b = not b

       if (random.randint(0,1)):

           c = True

           d = False

       else:

           d = True

           c = False

#Portanto são construções válidas:

# (a) A not is B.

# (b) A is B.

# (c) B not is A.

# (d) B is A.

&lt;/syntaxhighlight&gt;'''Objeções'''

# '''Isso significa que há apenas um número limitado de estados?''' Esse é um ponto central desse programa. A ideia é ancorar a gramática no que é biologicamente possível e numa máquina que diverge das máquinas de estado tradicionais.
# '''Faz asserções falseáveis no sentido do celébro filósofo Karl Popper?''' Primeiramente, a falseabilidade estritia já foi questionada por filósofos posteriores como Thomas Kuhn e Imre Lakatos&lt;ref&gt;{{Cite book|url=https://plato.stanford.edu/archives/fall2025/entrieshomas-kuhn/|title=Thomas Kuhn|last=Bird|first=Alexander|date=2025|publisher=Metaphysics Research Lab, Stanford University|editor-last=Zalta|editor-first=Edward N.|edition=Fall 2025|editor-last2=Nodelman|editor-first2=Uri}}&lt;/ref&gt;&lt;ref name=&quot;:0&quot; /&gt;. Cientistas notam que ele não entendeu a psicanálise como psicoterapia.&lt;ref&gt;{{Cite journal|last=Grant|first=Don C.|last2=Harari|first2=Edwin|date=2005-06|title=Psychoanalysis, science and the seductive theory of Karl Popper|url=https://pubmed.ncbi.nlm.nih.gov/15943645|journal=The Australian and New Zealand Journal of Psychiatry|volume=39|issue=6|pages=446–452|doi=10.1080/j.1440-1614.2005.01602.x|issn=0004-8674|pmid=15943645}}&lt;/ref&gt; Segundamente, a asserção central é que um jeito mais efetivo de considerar a gramática muito mais que o modelo de árvore tradicional e também de um jeito direcional: Sujeito -&gt; Verbo -&gt; Objeto em vez de Sentença { Sujeito Predicado { Verbo/Objeto.
# '''Como lidar com a recursão?''' A gramática binária poderia criar sentenças infinitas, dependo das condições necessárias. Por exemplo, é possível definir uma gramática de modo que há substantivos e preposições. É possível definir que a sentença começa com um substantivo e o que segue são interpolações de preposições com substantivos. Por exemplo, &quot;Pessoa do bem da comunidade de meio...&quot;. Essa seria uma maneira de definir uma gramática que atende a uma forma mais simples de recursão. A recursão não implica necessariamente em orações subordinadas.</text>
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__TOC__

== Overview ==
{{RoundBoxTop|theme=6}}
[[File:Clock and a jar of money labelled retirement.png|thumb|'''Figure 1'''. The retirement dilemma: balancing additional financial security from working longer against the value of time and freedom in retirement.]]

; More money, or more life?

Imagine you are nearing retirement age. You have always worked hard, built a secure career, and rarely thought about retirement. Then your family asks: “When do you actually want to retire?”

You realise you have never really considered it.

You start looking into your options, but the process feels overwhelming. You could keep working for several more years and build greater financial security, but what are those extra years worth? This dilemma reflects the trade-off between greater financial security and greater freedom in retirement (see Figure 1). More savings could mean more financial freedom later, but retiring sooner could mean more time now, to travel, spend time with family, pursue interests, and simply have control over your days.

You begin to wonder: Is working longer for more money worth postponing the freedom you could have today?

And perhaps the hardest question: Why didn't I start thinking about this sooner?

{{RoundBoxBottom}}

* Retirement is not simply an economic decision; people must weigh financial security against psychological, social, health, and emotional considerations.
* Work can provide income, social relationships, purpose, structure, autonomy, competence, and a valued identity, meaning that leaving employment may involve both gains and losses.
* Retirement decisions are therefore influenced by individual circumstances and motivations, making retirement timing highly variable between individuals (Feldman &amp; Beehr, 2011; Wang &amp; Huang, 2023).

{{RoundBoxTop|theme=6}}
'''Focus questions'''

* What factors influence people's decisions to retire?
* How do psychological needs and motivation influence retirement decisions?
* How does work identity affect the decision to retire?
* What emotional factors influence retirement?
* Why do some people continue working after they could retire?
{{RoundBoxBottom}}

== Factors influencing retirement decisions ==
* Retirement decisions are influenced by multiple interacting factors e.g., '''financial, health, work-related, social, and personal''', rather than age alone (Feldman &amp; Beehr, 2011; Wang &amp; Huang, 2023).

=== Financial security and retirement readiness ===
* Financial circumstances influence whether retirement is perceived as a realistic option, with insufficient financial resources potentially encouraging people to remain employed longer (Meng et al., 2020).
* Economic incentives can operate in both directions: financial security can enable retirement, while opportunities to accumulate additional income or savings can motivate continued employment (Meng et al., 2020).
* Retirement planning involves evaluating whether available financial resources will support an individual's desired lifestyle after leaving employment (Wang &amp; Huang, 2023).

=== Health and wellbeing ===
* Poor health can motivate earlier retirement, particularly when health problems make continued employment difficult (Meng et al., 2020).
* Physically or psychologically demanding work may become increasingly difficult to sustain as workers age (Stevens et al., 2022).
* Health and working conditions can therefore interact in retirement decision-making, with wellbeing potentially influencing both an individual's ability and desire to remain employed (Meng et al., 2020; Stevens et al., 2022).

=== Work and social circumstances ===
* High work demands, low decision authority, and limited recognition can contribute to preferences for earlier retirement or exit from paid employment (Carr et al., 2015).
* Autonomy, flexibility, supportive colleagues, and feeling valued can encourage older workers to remain employed (Stevens et al., 2022).
* Family circumstances, relationships, and opportunities for leisure may also influence when individuals want to leave employment (Meng et al., 2020).

== Psychological needs and motivation ==
* Psychological needs and the quality of an individual's motivation can influence whether continued employment or retirement is perceived as more personally rewarding (Halvari &amp; Olafsen, 2022; Ryan &amp; Deci, 2000).

=== Self-Determination Theory ===
* '''[[Self-determination theory|Self-Determination Theory]] (SDT)''' proposes that optimal motivation and wellbeing are supported by satisfaction of three basic psychological needs: '''autonomy, competence and relatedness''' (Ryan &amp; Deci, 2000).
* Work can provide opportunities to satisfy these needs through control and choice ('''autonomy'''), achievement and skill use ('''competence'''), and relationships with colleagues ('''relatedness''') (Ryan &amp; Deci, 2000).
* Retirement may alter how these psychological needs are satisfied, meaning that individuals may need to develop alternative activities and relationships outside employment that provide autonomy, competence and relatedness (Halvari &amp; Olafsen, 2022).

=== Intrinsic and extrinsic motivation ===

* '''[[Intrinsic motivation|Intrinsic]] or autonomous motivation''' may encourage continued employment when work remains enjoyable, interesting, personally meaningful or consistent with an individual's values (Halvari &amp; Olafsen, 2022).
* '''[[Extrinsic motivation|Extrinsic]] or controlled motivation''' may include financial rewards, expectations or external pressures that encourage an individual to continue working even when retirement is available (Ryan &amp; Deci, 2000; Halvari &amp; Olafsen, 2022).
* Research among workers aged 58–72 found that motivational profiles were associated with retirement-related outcomes, supporting the idea that the quality of work motivation can influence retirement intentions (Halvari &amp; Olafsen, 2022).

=== Goals and values in later life ===

* Retirement involves evaluating whether continued employment or retirement better supports an individual's current goals, priorities and desired lifestyle (Feldman &amp; Beehr, 2011).
* Goals may increasingly involve activities outside employment, such as leisure, relationships, travel or other personally meaningful pursuits, which can increase the attractiveness of retirement (Feldman &amp; Beehr, 2011).
* Retirement decisions can therefore involve a motivational trade-off between the valued outcomes provided by work and those expected from greater freedom and time outside employment (Feldman &amp; Beehr, 2011).

== Work identity and retirement ==
* Employment can provide identity, purpose, achievement, and belonging therefore retirement can represent a psychological and identity transition as well as the end of paid employment (Bordia et al., 2020).

=== Work as a source of identity ===
* Employment can become an important component of identity by providing purpose, status, achievement, social recognition and belonging (Bordia et al., 2020).
* Individuals differ in how strongly they identify with their occupational role, meaning that leaving work may represent a relatively minor change for some but a substantial identity disruption for others (Bordia et al., 2020).
* A strong work identity may influence retirement motivation because retiring involves relinquishing or transforming a role through which an individual has understood themselves for many years (Bordia et al., 2020).

=== Identity transition after retirement ===

* Retirement requires individuals to adjust from the role of worker to a different configuration of personal and social roles (Bordia et al., 2020).
* Maintaining multiple social identities outside employment may provide alternative sources of belonging, meaning and social support during retirement (Haslam et al., 2023).
* Activities such as family involvement, hobbies, volunteering and community participation may help retirees establish or strengthen meaningful identities outside their former occupation (Haslam et al., 2023).

{{RoundBoxTop|theme=6}}
'''If work has defined who I am, who will I be when I stop working?'''
{{RoundBoxBottom}}

== Emotions and retirement ==
* Retirement can involve both positive and negative emotional experiences, and emotions may be relevant both when anticipating retirement and when adjusting to life after work (Ryser &amp; Wernli, 2017; Ugwu et al., 2024).

=== Anticipated emotions ===
* Retirement decisions involve thinking about future outcomes, meaning that individuals may anticipate how they expect to feel if they retire or continue working (Feldman &amp; Beehr, 2011; Ryser &amp; Wernli, 2017).
* Retirement may be associated with anticipated positive emotions such as excitement, relief and freedom, but also negative emotions such as sadness, uncertainty or apprehension (Ryser &amp; Wernli, 2017).
* These anticipated emotional consequences may form part of the broader psychological evaluation of whether leaving employment is desirable (Feldman &amp; Beehr, 2011; Ryser &amp; Wernli, 2017).

=== Retirement anxiety ===

* Retirement can be associated with anxiety because it involves substantial changes to familiar routines, roles, goals and life circumstances (Ugwu et al., 2024).
* Uncertainty about life after work may make retirement psychologically challenging even when an individual is financially capable of retiring (Ugwu et al., 2024).
* Retirement anxiety is important because retirement represents not only the end of employment but also a transition into a substantially different stage of life (Ugwu et al., 2024).

=== Emotional adjustment to retirement ===

* Emotional responses to retirement vary considerably between individuals; retirement should therefore not be assumed to automatically improve or reduce wellbeing (Ryser &amp; Wernli, 2017).
* Adjustment may depend on the circumstances surrounding retirement and the individual's opportunities for satisfying activities and social participation after leaving work (Ryser &amp; Wernli, 2017).
* Successful emotional adjustment may involve establishing new routines, relationships, goals and meaningful activities that replace valued aspects of working life (Haslam et al., 2023; Ryser &amp; Wernli, 2017).

== Continuing to work after retirement age ==
* Reaching retirement eligibility does not necessarily eliminate motivation to work; financial, psychological, social, and work-related rewards can encourage older adults to remain employed (Halvari &amp; Olafsen, 2022; Stevens et al., 2022).

=== Why people choose to keep working ===
* Continuing employment can be a positive and motivated choice rather than simply a response to insufficient retirement savings (Halvari &amp; Olafsen, 2022).
* Older adults may remain employed because work provides enjoyment, purpose, achievement, identity and social relationships (Halvari &amp; Olafsen, 2022; Stevens et al., 2022).
* Financial motivation can coexist with psychological motivation, meaning that individuals may continue working for both additional financial security and the intrinsic benefits they receive from employment (Halvari &amp; Olafsen, 2022; Meng et al., 2020).

=== Job quality and continued work ===

* The quality of work can influence whether older employees want to remain in employment rather than retire (Carr et al., 2016; Stevens et al., 2022).
* Greater decision authority has been associated with preferences for later retirement, while high psychosocial demands have been associated with preferences for earlier retirement (Carr et al., 2015). 
* Flexibility, autonomy, supportive colleagues and feeling appreciated can make continued employment more attractive to older workers (Stevens et al., 2022).

=== Alternatives to full retirement ===

* Retirement does not necessarily involve an immediate transition from full-time employment to no paid work (Feldman &amp; Beehr, 2011).
* Some older adults may prefer '''part-time, reduced-hours or flexible employment''', allowing them to retain valued aspects of work while gaining greater control over their time (Stevens et al., 2022).
* Bridge employment can provide a gradual transition between career employment and full retirement, allowing individuals to maintain some benefits of working while adjusting to retirement (Wang et al., 2008).

== Conclusion ==

* Retirement is not motivated by age alone. Decisions about when to retire emerge from the interaction between financial circumstances, health, work conditions, psychological needs, social relationships, identity, and emotions.
* Motivation can operate in both directions: people may be motivated to retire to gain greater autonomy and time for valued activities, while they may also be motivated to continue working because employment provides income, competence, relationships, purpose, or identity.
* Retirement should therefore be understood as a '''psychological transition as well as an economic decision'''. Planning for retirement involves considering not only whether a person can afford to stop working, but also what will give them meaning, social connection, autonomy, and purpose afterwards.
* Ultimately, the most desirable retirement timing is unlikely to be the same for everyone. Understanding individual motivations and anticipated emotional experiences can help people make retirement decisions that better align with their goals and values.

== Learning feature ==
{{RoundBoxTop|theme=6}}
'''Two Paths to Retirement'''

Imagine two people of the same age who have enough financial resources to retire.

Person A finds their work highly meaningful, enjoys their colleagues, and values the sense of achievement their career provides. They choose to continue working part-time.

Person B enjoys their work but increasingly values spending time with family, travelling, and pursuing interests outside employment. They choose to retire.

Learning purpose: This scenario will demonstrate that financial readiness does not determine retirement motivation on its own. Differences in autonomy, relatedness, competence, work identity, goals, and anticipated emotions can lead people with similar financial circumstances to make different retirement decisions.
{{RoundBoxBottom}}

=== Quiz Question ===
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement best explains why two people with similar financial circumstances might make different retirement decisions?
|type=&quot;()&quot;}

- Retirement decisions are determined primarily by age.
+ People have different psychological needs, identities, goals, and emotional expectations.
- People who continue working are necessarily more motivated.
- People who retire earlier necessarily have lower job satisfaction.
&lt;/quiz&gt;

== See also ==

* [[Extrinsic motivation]] (Wikiversity)
* [[Intrinsic motivation]] (Wikiversity)
* [[Self-determination theory|Self-Determination Theory]] (Wikiversity)
* [[wikipedia:Self-determination_theory|Self-Determination Theory]] (Wikipedia)

== References ==
{{Hanging indent|1=

Bordia, P., Read, S., &amp; Bordia, S. (2020). Retiring: Role identity processes in retirement transition. Journal of Organizational Behavior, 41(5), 445–460. https://doi.org/10.1002/job.2438

Carr, E., Hagger-Johnson, G., Head, J., Shelton, N., Stafford, M., Stansfeld, S., &amp; Zaninotto, P. (2015). Working conditions as predictors of retirement intentions and exit from paid employment: A 10-year follow-up of the english longitudinal study of ageing. European Journal of Ageing, 13(1), 39–48. https://doi.org/10.1007/s10433-015-0357-9

Feldman, D. C., &amp; Beehr, T. A. (2011). A three-phase model of retirement decision making. American Psychologist, 66(3), 193–203. https://doi.org/10.1037/a0022153

Halvari, H., &amp; Olafsen, A. H. (2022). Will they stay or will they go? Motivational profiles, retirement-related correlates, and retirement intentions among 58–72-year-old workers. Frontiers in Psychology, 13, 807752. https://doi.org/10.3389/fpsyg.2022.807752

Haslam, C., Lam, B. C. P., Ghafoori, E., Steffens, N. K., Haslam, S. A., Bentley, S. V., Cruwys, T., &amp; La Rue, C. J. (2023). A longitudinal examination of the role of social identity in supporting health and well-being in retirement. Psychology and Aging, 38(7), 615–626. https://doi.org/10.1037/pag0000757

Lawrence Ejike Ugwu, Wojujutari Kenni Ajele, &amp; Erhabor Sunday Idemudia. (2024). Paradox of life after work: A systematic review and meta-analysis on retirement anxiety and life satisfaction. PLOS Global Public Health, 4(4), e0003074–e0003074. https://doi.org/10.1371/journal.pgph.0003074

Meng, A., Sundstrup, E., &amp; Andersen, L. L. (2020). Factors contributing to retirement decisions in denmark: Comparing employees who expect to retire before, at, and after the state pension age. International Journal of Environmental Research and Public Health, 17(9), 3338. https://doi.org/10.3390/ijerph17093338

Ryan, R. M., &amp; Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55(1), 68–78. https://doi.org/10.1037/0003-066X.55.1.68

Ryser, V.-A., &amp; Wernli, B. (2017). How does transitioning into retirement impact the individual emotional system? Evidence from the swiss context. Advances in Life Course Research, 32, 42–54. https://doi.org/10.1016/j.alcr.2016.08.001

Stevens, M. J., Barker, M., Dennison, E., Harris, E. C., Linaker, C., Weller, S., &amp; Walker-Bone, K. (2022). Recent UK retirees’ views about the work-related factors which influenced their decision to retire: A qualitative study within the Health and Employment After Fifty (HEAF) cohort. BMC Public Health, 22, 116. https://doi.org/10.1186/s12889-022-12541-1

Ugwu, L. E., Ajele, W. K., &amp; Idemudia, E. S. (2024). Paradox of life after work: A systematic review and meta-analysis on retirement anxiety and life satisfaction. PLOS Global Public Health, 4(4), e0003074. https://doi.org/10.1371/journal.pgph.0003074

Wang, M., &amp; Huang, Z. (2023). A contemporary review of employee retirement. Current Opinion in Psychology, 55, 101749–101749. https://doi.org/10.1016/j.copsyc.2023.101749

Wang, M., Zhan, Y., Liu, S., &amp; Shultz, K. S. (2008). Antecedents of bridge employment: A longitudinal investigation. Journal of Applied Psychology, 93(4), 818–830. https://doi.org/10.1037/0021-9010.93.4.818

}}

== External links ==

* '''Age Pension''' ([https://www.servicesaustralia.gov.au/age-pension? Services Australia])
** Information about Age Pension eligibility, income and assets tests, and how to prepare for and claim the Age Pension.
* '''Older Australians: Employment and work''' ([https://www.aihw.gov.au/reports/older-people/older-australians/contents/employment-and-work? Australian Institute of Health and Welfare])
** Australian data and information about older people's workforce participation, employment, and retirement.
* '''Plan for your retirement''' ([https://moneysmart.gov.au/plan-for-your-retirement? Moneysmart])
** An Australian Government resource providing practical information about retirement planning, including financial preparation, retirement timing, lifestyle goals, health, and options for continuing to work.

* '''Retirement Standard''' ([https://www.superannuation.asn.au/consumers/retirement-standard/ Association of Superannuation Funds of Australia])
** Australia’s benchmark guide helps you determine how much money you need for retirement, empowering you to make informed decisions about your future after work.

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      <text bytes="13018" sha1="dn5nanff3pbyc1vi6nod2byr2s3vejo" xml:space="preserve">{{title|Need to love and be loved:&lt;br&gt;How does the desire to give and receive love influence motivation?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=12}}
[[File:Love heart.jpg|thumb|150px|'''Figure 1'''. Hands holding out a heart.]]

'''Understanding love in pop culture'''

''Do you ever wonder how the media portrays love-driven behaviours?''

Filmmakers and game developers intentionally create love-driven character stories as it floods our brain with oxytocin (the &quot;bonding and empathy&quot; hormone). In turn, this hormone enhances our viewing experience, making films and video games much more memorable (Zak, 2015).

In the 2026 psychological horror film the [https://en.wikipedia.org/wiki/Backrooms_(film) Backrooms] {{ic|Use an internal link as shown in Tutorial 2}}, the deuteragonist, Mary, investigates the disappearance of her patient, Clark. She visits his workplace, ''Cap'n Clark's Ottoman Empire'', and unknowingly stumbles across the backrooms of the establishment. Mary's decision to search for Clark requires considerable risk and effort. Her behaviour raises the question of a person might pursue another person's wellbeing, even when doing so conflicts with their own safety. A more positive example of giving and receiving love is portrayed in the popular 2013 video game [https://en.wikipedia.org/wiki/The_Last_of_Us The Last of Us]. The game centres around a hardened smuggler named Joel who must escort a teenage girl named Ellie across a post-apocalyptic United States. As they spend more time together, their relationships develops into a close emotional bond resembling that of a parent and child.
{{RoundBoxBottom}}

The Overview section should provide:
# '''Scenario''': (see above) ✅
# '''Explanation of the problem, issue, or topic''': Briefly explain the problem, why it is important, and outline how psychological science can help ✅
#* '''Motivation''': First, explain the importance of motivation and how it is a psychological process that influences direction, intensity, and the persistence of a behaviour. 
#* '''Love and Belonging''': Follow with introducing the concepts of love and belonging. Love and belonging is needs involving affection, acceptance, and connection. Love is not limited to romantic relationships (THIS WILL BE EMPHASISED).
#* '''Interpersonal Connection''': This term can bridge the above concepts to the theories. The need for meaningful connection is considered in several psychological perspectives: Maslow's Hierarchy of Needs, Self-determination theory, and Attachment theory. 
#* '''Outline the aim of the chapter'''; (i.e. this chapter develops the readers{{g}} understanding of the desire to give and receive love, and how motivation plays an important role in influencing said behaviours. This chapter will draw on a breath of psychological research, including key concepts and terms, and draw upon different psychological theories related to understanding love. This is to provide the reader a comprehensive but applicable direction for knowledge.

'''''DRAFT OVERVIEW:'''''

Love is a complex interpersonal experience that can influence how people think, feel, and behave. In psychology, motivation refers to the processes that give behaviour direction, intensity, and persistence. The desire to love and be loved can therefore be understood as a source of motivation. Motivation can influence people to initiate and invest in meaningful relationships. Love and belonging involve the desire for affection and acceptance, while interpersonal connection refers to the emotional and social bonds people develop with others.

Several psychological theories provide different perspectives for understanding this relationship between love and motivation. Maslow's hierarchy of needs conceptualises love and belonging as fundamental humans needs, while basic psychological needs theory identifies relatedness, or feeling connected to and cared for by others, as a fundamental psychological need. Attachment theory provides another framework for understanding how close emotional bonds can influence behaviour. Particularly, caregiving and responses to separation or threat. 

Recommended length: 180-330 words. 
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box (see below) ✅

{{RoundBoxTop|theme=12}}'''Focus questions'''

{{ic|Use bullet points as shown in Tutorial 2}}

1. Why is the need to love and be loved considered an important source of human motivation?

2. How do psychological theories explain the motivation to form and maintain meaningful relationships?

3. How does the desire to give and receive love influence behaviours such as affiliation, caregiving, and prosocial behaviour?

4. When can the motivation to obtain love and belonging conflict with other psychological needs or become maladaptive?

{{RoundBoxBottom}}

==1. Love and Belonging as Human Motivation==
{{expand}}

====== 1.1 The Need to Belong ======
[[File:Human–canine friendship - girl hugging her dog tightly at golden hour in Laos.jpg|thumb|'''Figure 2'''. A companionship that goes beyond.]]

Love and interpersonal relationships can influence motivation by directing people towards social connection. In other words, forming and maintaining relationships. Humans do not simply interact with others because social interaction is available. Baumeister and Leary (1995)'s [https://www.researchgate.net/profile/Mark-Leary-2/publication/15420847_The_Need_to_Belong_Desire_for_Interpersonal_Attachments_as_a_Fundamental_Human_Motivation/links/5b647053aca272e3b6af9211/The-Need-to-Belong-Desire-for-Interpersonal-Attachments-as-a-Fundamental-Human-Motivation.pdf Theory of Belonging] reviewed evidence supporting the belongingness hypothesis and concluded that people have a pervasive desire to form social attachments.  

People readily develop relationships with others, including under adverse circumstances, and tend to resist the loss or dissolution of existing social bonds even when maintaining them provides no obvious practical benefit. This suggests that interpersonal relationships can function as motivation goals in their own right rather than simply serving other purposes. The importance of belonging can be seen in the psychological consequences associated with social connection and disconnection. Rather than being simply a preference for social interaction, belongingness can influence emotional and cognitive processes. This explains why people may invest considerable time and effort into developing and maintaining relationships. The desire for connection can also become particularly influential when an individual's relationship is perceived as a threat or when dependence on another person becomes excessive. For example, Guan et al., (2015) examined 80 couples longitudinally and found that love addiction was positively associated with emotional dependence and negatively associated with attachment avoidance. Their findings suggest that the motivation to maintain close romantic relationships can interact with attachment patterns and interpersonal dependence.  

====== 1.2 Love, Connection, And Relatedness ======
Belongingness refers to the human emotion to be accepted as a member of a group. Whether family, friends, co-workers, a religion, or something else. People have an inherent desire to belong.   

[[File:20110702-31-Ilana_tim_and_becca_hug_(5972579616).jpg|thumb|225x225px|'''Figure 3'''. Friends hugging.]]

Love can therefore be one way through which the broader need for interpersonal connection is experienced and expressed. Qadir et al., (2026) explored occupational therapists' experiences of belongingness within interprofessional clinical environments and found colleagues and patients contributed to participants' sense of belonging. Conversely, organisational barriers, discrimination and difficulties establishing a professional identity could undermine this sense of connection. The study also illustrates how belongingness and motivation can reinforce one another. Participants identified achievement, professional development and appreciation from others as important sources of motivation, while supportive and welcoming interactions contributed to their sense of belonging. This suggests that interpersonal environments can both motivate individuals to participate and develop the social acceptance that contributes to belongingness.  

{{RoundBoxTop|theme=12}}'''Think About It: What's On Your Phone Lock Screen?'''

[[File:IPhone 17e Soft Pink Model - 1.jpg|thumb|150px|'''Figure 4'''. Pink Apple iPhone.]]

''Imagine you could only have one person on your phone's lock screen for the next year. Who would you choose, and why them?''

A lock screen is a practical feature of a phone, yet the image we choose to display can carry considerable emotional meaning. People may choose photographs of partners, family members, friends, or other significant individuals, turning an everyday object into a reminder of an important relationship. Conroy and Fraley (2026) investigated whether photographs of romantic partners could function as everyday attachment-related cues. Their findings suggested that lock screens are embedded with psychological processes involved in close relationships. Simply put, people like having photos of their partners.{{RoundBoxBottom}}

== 2. Theories Explaining Love ==
{{expand}}

====== 2.1 Maslow's Hierarchy of Needs ======
Maslow (1943) proposed that human motivation is influenced by a hierarchy of needs. Beginning with physiological needs and safety, then love and belonging, and finally esteem and self-actualisation. Within this framework, love and belonging involve the desire for affection and acceptance. When these needs are insufficiently satisfied, individuals may become motivated to seek relationships and social connection.

====== 2.2 Self-Determination Theory ======
People are centrally concerned with motivation, how to move themselves or others to act (Ryan &amp; Deci, 2000).

== 3. How Love Influences Motivated Behaviour ==
What does this motivation actually make people do?

==Conclusion==
* Draw back to the Overview; discuss the terms, and how those concepts link to the theories discussed.
* Love and interpersonal relationships can influence motivation by directing people towards social connections.
* Belongingness can influence emotional and cognitive processes, explaining why individuals invest time and effort in maintaining their relationships despite interfering with other life goals.
* Draw on the comparison between Maslow's theory, self-determination theory, and attachment theory. 
* Potentially answer the focus questions, or at least, reference them once more here.
* Recommended length: 150 to 330 words

==See also==
* [[wikipedia:Belongingness|Belongingness]] (Wikipedia)
* [[Motivation and emotion/Book/2025/Love styles and relationship satisfaction|Love styles and relationship satisfaction]] (Book chapter, 2025)
* [[wikipedia:Need_theory|Need theory]] (Wikipedia)

==References==
{{Hanging indent|1=

Baumeister, R. F., &amp; Leary, M. R. (1995). The need to belong: Desire for interpersonal attachments as a fundamental human motivation. ''Psychological Bulletin, 117''(3), 497-529. https://doi.org/10.1037/0033-2909.117.3.497

Conroy, K. E., &amp; Fraley, R. C. (2026). Security priming in everyday life: How do symbols of close others support attachment in adulthood? ''Journal of Personality and Social Psychology, 130''(6), 1229-1247. https://doi.org/10.1037/pspp0000590

Guan, C., Wang, J., Zhang, L. et al. (2025). A longitudinal network of the relationship between love addiction, insecure attachment patterns, and interpersonal dependence. ''BMC Psychol'' 13, 330. https://doi.org/10.1186/s40359-025-02605-3

Maslow, A. H. (1943). A theory of human motivation. ''Psychological Review, 50''(4), 370-396. https://doi.org/10.1037/h0054346

Ryan, R. M., &amp; Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. ''American Psychologist, 55''(1), 68-79. https://doi.org/10.1037/0003-066x.55.1.68

Qadir, N. A., &amp; Sethi, A. (2026). Factors affecting belongingness of Occupational Therapists in an Interprofessional Clinical Environment. ''Pakistan Journal of Medical Sciences, 42''(5), 1100-1104. https://doi.org/10.12669/pjms.42.5.13539

Zak, P. J (2015). Why inspiring stories make us react: the neuroscience of narrative. ''Cerebrum: the Dana forum on brain science, 2015'', 2.

}}

==External links==

* [https://www.unitedfamilies.org/parents/child-development/love-the-greatest-motivation-to-change/ Love: The Greatest Motivation to Change] (unitiedfamilies.org)
* [https://www.youtube.com/watch?v=4BvlBoZsV_I The Key to Writing Criminally Good Relationships | Video Essay] (YouTube)
* [https://open.spotify.com/track/42WQZAqPJOlLvg2Jr6lv8K You're My Best Friend - Queen] (Spotify)

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&lt;div align=center&gt;Edit the title and sub-title to match the wording (and casing) in the [[Motivation and emotion/Book/2025|2026 list of topics]].&lt;br&gt;[[Motivation and emotion/About/Staff|Seek approval]] for any changes.&lt;br&gt;Do not include your name (authorship is as per [[Special:History/{{PAGENAME}}|the page history]]).&lt;/div&gt;
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==Overview==
{{RoundBoxTop|theme=3}}
[[File:Botellón at San Giovanni.jpg|right|thumb|150px|'''Figure 1'''. People part taking in sensation-seeking]]

;Scenario

Jaemin often feels dull and struggles to maintain contentment by doing boring daily activities such as working and chores at home. 

On the weekend Jaemin tends to feel the happiest when going out partying, doing social smoking and getting laid. 
He feels alive but soon after the weekend is over falls back into his dull life just getting by till the next weekend to do it over again. 


{{RoundBoxBottom}}

Dopamine and Sensation-Seeking have a difficult relationship, especially in very different parts of the brain. As confusing as these two are together, they are essential to many decisions we make as people. From disorders such as ADHD and drug use attempting to search for something more, to individuals who may speed faster to feel the wind and a rush in their body. Both dopamine and sensation-seeking connect together neurologically to make these moments happen. But the question is why does this happen? and what is this relationship which happens in these moments? 

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What is the relationship between sensation-seeking and dopamine?
* What is the brains part in creating dopamine and sensation-seeking? 
* Is there a reason why dopamine and sensation-seeking connect together?
* How has the neurobiological relationship evolved over time?


{{RoundBoxBottom}}

==Dopamine And Sensation-Seeking?==

=== What Is Dopamine? ===

=== What is Sensation-Seeking? ===
* What is dopamine?.
* What is sensation-seeking?.
* Why they are important?

== The Neurobiological Mechanics ==

=== Basal Ganglia ===
Basal Ganglia consists of interconnected nuclei and works by receiving action plans, and executing plans (Reeves, 2024).

=== Ventral Striatum ===

=== Ventral Tegmental (VTA) ===
Ventral Tegmental is the main area in the Basal Ganglia which produces dopamine also known as (VTA) (Reeves, 2024).

=== Nucleus Accumbens (NAc) ===
The Nucleus Accumbens holds the core reward centre and processes pleasure within the brain and reinforcement (Reeves, 2024)

* What parts of the brain create dopamine and sensation-seeking?
* How do they connect and interact with each other?

== The Relationship Of Dopamine And Sensation-Seeking ==
* How does Dopamine and Sensation-seeking work outside of the body?
* How does this effect individuals with high risk-taking?
* How has the relationship evolved over time?
[[File:Dopamine structure.svg|thumb|140x140px|'''Figure 2'''. The chemical compound of dopamine.]]
;
&lt;quiz display=&quot;simple&quot;&gt;
{can sensation seeking happen without releasing dopamine?:
|type=&quot;( )&quot;}
- True
+ False

&lt;/quiz&gt;

==Conclusion==
* There is a neurobiological relationship between dopamine and sensation-seeking.
* The brain does have main areas which affects dopamine and sensation-seeking.
* The neurobiological relationship has evolved over time with research.

{{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==
* [https://w.wiki/Tqc9 Dopamine receptor D4] (Wikipedia)
* [[Motivation and emotion/Book/2024/Dopamine and decision making]] (Wikiversity)
* [[Motivation and emotion/Book/2014/Dopamine and motivation]] (Wikiversity)
* [[Motivation and emotion/Book/2011/Sensation seeking]] (Wikiversity)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; 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 &amp; Row.

Kauê Machado Costa, Geoffrey Schoenbaum, Dopamine, Current Biology, Volume 32, Issue 15, 2022, Pages R817-R824, ISSN 0960-9822, https://doi.org/10.1016/j.cub.2022.06.060

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://www.youtube.com/watch?v=2po3w5oFygY Dopamine System, Craving &amp; Pursuit Explained] (Youtube)
* [https://www.youtube.com/watch?v=EJbbMasBSGY&amp;t=192s Just For The Thrill Of It: An Inside Look At Sensation Seeking] (Youtube)
* [https://www.youtube.com/watch?v=Wa8_nLwQIpg 2-Minute Neuroscience: Dopamine] (Youtube)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
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&lt;div align=center&gt;Edit the title and sub-title to match the wording (and casing) in the [[Motivation and emotion/Book/2025|2026 list of topics]].&lt;br&gt;[[Motivation and emotion/About/Staff|Seek approval]] for any changes.&lt;br&gt;Do not include your name (authorship is as per [[Special:History/{{PAGENAME}}|the page history]]).&lt;/div&gt;
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==Overview==
{{RoundBoxTop|theme=3}}
[[File:Botellón at San Giovanni.jpg|right|thumb|150px|'''Figure 1'''. People part taking in sensation-seeking]]

;Scenario

Jaemin often feels dull and struggles to maintain contentment by doing boring daily activities such as working and chores at home. 

On the weekend Jaemin tends to feel the happiest when going out partying, doing social smoking and getting laid. 
He feels alive but soon after the weekend is over falls back into his dull life just getting by till the next weekend to do it over again. 


{{RoundBoxBottom}}

Dopamine and Sensation-Seeking have a difficult relationship, especially in very different parts of the brain. As confusing as these two are together, they are essential to many decisions we make as people. From disorders such as ADHD and drug use attempting to search for something more, to individuals who may speed faster to feel the wind and a rush in their body. Both dopamine and sensation-seeking connect together neurologically to make these moments happen. But the question is why does this happen? and what is this relationship which happens in these moments? 

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What is the relationship between sensation-seeking and dopamine?
* What is the brains part in creating dopamine and sensation-seeking? 
* Is there a reason why dopamine and sensation-seeking connect together?
* How has the neurobiological relationship evolved over time?


{{RoundBoxBottom}}

==Dopamine And Sensation-Seeking?==

=== What Is Dopamine? ===

=== What is Sensation-Seeking? ===
* What is dopamine?.
* What is sensation-seeking?.
* Why they are important?

== The Neurobiological Mechanics ==

=== Basal Ganglia ===
Basal Ganglia consists of interconnected nuclei and works by receiving action plans, and executing plans (Reeves, 2024).

=== Ventral Striatum ===

=== Ventral Tegmental (VTA) ===
Ventral Tegmental is the main area in the Basal Ganglia which produces dopamine also known as (VTA), (Reeves, 2024).

=== Nucleus Accumbens (NAc) ===
The Nucleus Accumbens holds the core reward centre and processes pleasure within the brain and reinforcement (Reeves, 2024).

* What parts of the brain create dopamine and sensation-seeking?
* How do they connect and interact with each other?

== The Relationship Of Dopamine And Sensation-Seeking ==
* How does Dopamine and Sensation-seeking work outside of the body?
* How does this effect individuals with high risk-taking?
* How has the relationship evolved over time?
[[File:Dopamine structure.svg|thumb|140x140px|'''Figure 2'''. The chemical compound of dopamine.]]
;
&lt;quiz display=&quot;simple&quot;&gt;
{can sensation seeking happen without releasing dopamine?:
|type=&quot;( )&quot;}
- True
+ False

&lt;/quiz&gt;

==Conclusion==
* There is a neurobiological relationship between dopamine and sensation-seeking.
* The brain does have main areas which affects dopamine and sensation-seeking.
* The neurobiological relationship has evolved over time with research.

{{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==
* [https://w.wiki/Tqc9 Dopamine receptor D4] (Wikipedia)
* [[Motivation and emotion/Book/2024/Dopamine and decision making]] (Wikiversity)
* [[Motivation and emotion/Book/2014/Dopamine and motivation]] (Wikiversity)
* [[Motivation and emotion/Book/2011/Sensation seeking]] (Wikiversity)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; 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 &amp; Row.

Kauê Machado Costa, Geoffrey Schoenbaum, Dopamine, Current Biology, Volume 32, Issue 15, 2022, Pages R817-R824, ISSN 0960-9822, https://doi.org/10.1016/j.cub.2022.06.060

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://www.youtube.com/watch?v=2po3w5oFygY Dopamine System, Craving &amp; Pursuit Explained] (Youtube)
* [https://www.youtube.com/watch?v=EJbbMasBSGY&amp;t=192s Just For The Thrill Of It: An Inside Look At Sensation Seeking] (Youtube)
* [https://www.youtube.com/watch?v=Wa8_nLwQIpg 2-Minute Neuroscience: Dopamine] (Youtube)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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__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 result.]]

'''Scenario'''

Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.
* '''What are emotional intelligence and emotional wellbeing?'''  (Define emotional intelligence and emotional wellbeing. Explain that emotional intelligence generally involves recognising, understanding, using, and managing emotions. Explain emotional wellbeing and clarify that wellbeing does not simply mean always experiencing positive emotions. Introduce the idea that there are different models of emotional intelligence.)  '''Research needed:''' A foundational source defining emotional intelligence, such as Mayer and Salovey, plus a reliable source defining emotional/subjective wellbeing (Givon et al., 2020).
* '''What is the relationship between emotional intelligence and emotional wellbeing?'''  (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

'''Figure 2. The four-branch model of emotional intelligence'''

Create a simple diagram showing:

Perceiving emotions → Using emotions → Understanding emotions → Managing emotions

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===

* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Nadler, R., Carswell , J., &amp; 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/

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819


Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://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]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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__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 result.]]

'''Scenario'''

Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.
* '''What are emotional intelligence and emotional wellbeing?''' 
* Emotional intelligence is the ability to identify, understand, use, and manage emotions. Salovey and Mayer first defined EI as a set of skills that help people accurately recognise and express emotions in themselves and others, regulate emotions effectively, and use emotions to motivate, plan, and achieve goals in life (Salovey &amp; R Caruso, 2008). Mayer and Salovey later developed a four-branch model that includes four related abilities: perceiving, using, understanding, and managing emotions (Salovey &amp; Grewal, 2005)
* (Define emotional intelligence and emotional wellbeing. Explain that emotional intelligence generally involves recognising, understanding, using, and managing emotions. Explain emotional wellbeing and clarify that wellbeing does not simply mean always experiencing positive emotions. Introduce the idea that there are different models of emotional intelligence.)  '''Research needed:''' A foundational source defining emotional intelligence, such as Mayer and Salovey, plus a reliable source defining emotional/subjective wellbeing (Givon et al., 2020).
* '''What is the relationship between emotional intelligence and emotional wellbeing?'''  (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

'''Figure 2. The four-branch model of emotional intelligence'''

Create a simple diagram showing:

Perceiving emotions → Using emotions → Understanding emotions → Managing emotions

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===

* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Nadler, R., Carswell , J., &amp; 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/

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/yp64q2ftjn?route=details

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819


Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://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]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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__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 result.]]

'''Scenario'''

Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.
* '''What are emotional intelligence and emotional wellbeing?''' 
* (Define emotional intelligence and emotional wellbeing. Explain that emotional intelligence generally involves recognising, understanding, using, and managing emotions. Explain emotional wellbeing and clarify that wellbeing does not simply mean always experiencing positive emotions. Introduce the idea that there are different models of emotional intelligence.)  '''Research needed:''' A foundational source defining emotional intelligence, such as Mayer and Salovey, plus a reliable source defining emotional/subjective wellbeing 
Emotional intelligence (EI) means being able to recognise, understand, use, and manage emotions (Givon et al., 2020). There are three main ways to think about EI: skill-based models see it as a type of intelligence, trait-based models view it as emotion-related self-perceptions, and mixed models combine abilities with personality traits (Luna et al., 2021). The four-branch model by Mayer and Salovey, which is the most influential ability account, defines emotional intelligence as perceiving, using, understanding, and managing emotions (Salovey &amp; Grewal, 2005). ). In contrast, trait EI theory views the concept as emotional self-efficacy: a person's natural tendency to notice and manage their own emotions. 

Emotional wellbeing concerns how people feel and how they judge the quality of their emotional lives. Researchers usually study it as part of subjective wellbeing (SWB), which examines how people think and feel about their lives (Villanueva et al., 2020). Some recent research suggests that emotional wellbeing is its own type of wellbeing. It includes positive feelings and life satisfaction, but it is not the same as mental health or simply the absence of illness (Park et al., 2022). Wellbeing does not mean always feeling good. Instead, it is about having more pleasant than unpleasant feelings and being satisfied with life. These ideas are closely related. Studies show a moderate positive link between emotional intelligence (EI) and subjective wellbeing (r=.32r=.32) (Xu et al., 2021). Another large review of over 3,000 studies and more than one million people found a strong connection between EI and human flourishing (r=.28r=.28) (Robinson, 2026). However, this link is not always the same. It varies by EI model and by whether wellbeing is measured by feelings or life satisfaction (Llamas-Díaz et al., 2022). This supports the idea that emotional intelligence is complex, and different emotional skills may affect emotional wellbeing in different ways.

* '''What is the relationship between emotional intelligence and emotional wellbeing?'''  (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

'''Figure 2. The four-branch model of emotional intelligence'''

Create a simple diagram showing:

Perceiving emotions → Using emotions → Understanding emotions → Managing emotions

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===

* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; 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/

Park, C. L., Kubzansky, L. D., Chafouleas, S. M., Davidson, R. J., Keltner, D., Parsafar, P., Conwell, Y., Martin, M. Y., Hanmer, J., &amp; Wang, K. H. (2022). Emotional well-being: What it is and why it matters. Affective Science, 4(1). https://doi.org/10.1007/s42761-022-00163-0

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://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]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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__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 result.]]

'''Scenario'''

Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.
* '''What are emotional intelligence and emotional wellbeing?''' 
* (Define emotional intelligence and emotional wellbeing. Explain that emotional intelligence generally involves recognising, understanding, using, and managing emotions. Explain emotional wellbeing and clarify that wellbeing does not simply mean always experiencing positive emotions. Introduce the idea that there are different models of emotional intelligence.)  '''Research needed:''' A foundational source defining emotional intelligence, such as Mayer and Salovey, plus a reliable source defining emotional/subjective wellbeing 
Emotional intelligence (EI) refers to the ability to recognise, understand, use, and manage emotions (Givon et al., 2020). There are several approaches to understanding EI. Ability models view EI as a form of intelligence, trait models focus on people's perceptions of their own emotional abilities, and mixed models combine emotional abilities with personality characteristics (Luna et al., 2021). One of the most influential ability approaches is Mayer and Salovey's four-branch model. This model describes EI through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions (Salovey &amp; Grewal, 2005). In comparison, trait EI focuses more on how people perceive their own emotional abilities and tendencies. 

Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. It is commonly examined as part of subjective wellbeing, which considers both how people feel and how satisfied they are with their lives (Villanueva et al., 2020). Emotional wellbeing is not simply the absence of mental illness, nor does it mean feeling happy or positive all the time (Park et al., 2022). Negative emotions such as sadness, anger, anxiety, and disappointment are a normal part of life. Instead, emotional wellbeing involves being able to experience and respond to different emotions while maintaining positive emotional functioning and overall life satisfaction.

* '''What is the relationship between emotional intelligence and emotional wellbeing?'''
Research generally suggests that emotional intelligence is positively associated with wellbeing. Xu et al. (2021) found a moderate positive relationship between EI and subjective wellbeing (''r'' = .32), suggesting that people with higher EI also tend to report greater wellbeing. Similarly,  Robinson &amp; Zell (2026) reported a positive association between EI and human flourishing (''r'' = .28). However, the relationship between EI and wellbeing is not necessarily the same across all studies. Findings may differ depending on the model of emotional intelligence being examined and whether wellbeing is measured through emotional experiences, life satisfaction, or other indicators (Llamas-Díaz et al., 2022). This suggests that the relationship between emotional intelligence and emotional wellbeing is complex, with different emotional abilities potentially contributing to wellbeing in different ways.
* (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

'''Figure 2. The four-branch model of emotional intelligence'''

Create a simple diagram showing:

Perceiving emotions → Using emotions → Understanding emotions → Managing emotions

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===

* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; 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/

Park, C. L., Kubzansky, L. D., Chafouleas, S. M., Davidson, R. J., Keltner, D., Parsafar, P., Conwell, Y., Martin, M. Y., Hanmer, J., &amp; Wang, K. H. (2022). Emotional well-being: What it is and why it matters. Affective Science, 4(1). https://doi.org/10.1007/s42761-022-00163-0

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://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]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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__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 result.]]

'''Scenario'''

Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.
* '''What are emotional intelligence and emotional wellbeing?''' 
* (Define emotional intelligence and emotional wellbeing. Explain that emotional intelligence generally involves recognising, understanding, using, and managing emotions. Explain emotional wellbeing and clarify that wellbeing does not simply mean always experiencing positive emotions. Introduce the idea that there are different models of emotional intelligence.)  '''Research needed:''' A foundational source defining emotional intelligence, such as Mayer and Salovey, plus a reliable source defining emotional/subjective wellbeing 
Emotional intelligence (EI) refers to the ability to recognise, understand, use, and manage emotions (Givon et al., 2020). There are several approaches to understanding EI. Ability models view EI as a form of intelligence, trait models focus on people's perceptions of their own emotional abilities, and mixed models combine emotional abilities with personality characteristics (Luna et al., 2021). One of the most influential ability approaches is Mayer and Salovey's four-branch model. This model describes EI through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions (Salovey &amp; Grewal, 2005). In comparison, trait EI focuses more on how people perceive their own emotional abilities and tendencies. 

Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. It is commonly examined as part of subjective wellbeing, which considers both how people feel and how satisfied they are with their lives (Villanueva et al., 2020). Emotional wellbeing is not simply the absence of mental illness, nor does it mean feeling happy or positive all the time (Park et al., 2022). Negative emotions such as sadness, anger, anxiety, and disappointment are a normal part of life. Instead, emotional wellbeing involves being able to experience and respond to different emotions while maintaining positive emotional functioning and overall life satisfaction.

* '''What is the relationship between emotional intelligence and emotional wellbeing?'''
Research generally suggests that emotional intelligence is positively associated with wellbeing. Xu et al. (2021) found a moderate positive relationship between EI and subjective wellbeing (''r'' = .32), suggesting that people with higher EI also tend to report greater wellbeing. Similarly, a large second-order meta-analysis of more than 3,000 studies and over one million participants found a positive association between emotional intelligence and human flourishing (''r'' = .28) (Robinson &amp; Zell, 2026). However, the strength of this relationship varied across studies and was stronger when emotional intelligence and flourishing were measured using self-report measures. Findings may also differ depending on the model of emotional intelligence being examined and whether wellbeing is measured through emotional experiences, life satisfaction, or other indicators (Llamas-Díaz et al., 2022). Overall, these findings suggest that emotional intelligence is related to greater wellbeing, although the strength of this relationship may depend on how both concepts are defined and measured.
*

* (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

'''Figure 2. The four-branch model of emotional intelligence'''

Create a simple diagram showing:

Perceiving emotions → Using emotions → Understanding emotions → Managing emotions

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===

* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; 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/

Park, C. L., Kubzansky, L. D., Chafouleas, S. M., Davidson, R. J., Keltner, D., Parsafar, P., Conwell, Y., Martin, M. Y., Hanmer, J., &amp; Wang, K. H. (2022). Emotional well-being: What it is and why it matters. Affective Science, 4(1). https://doi.org/10.1007/s42761-022-00163-0

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

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* [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]

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__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 result.]]

'''Scenario'''

Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.
* '''What are emotional intelligence and emotional wellbeing?''' 
* (Define emotional intelligence and emotional wellbeing. Explain that emotional intelligence generally involves recognising, understanding, using, and managing emotions. Explain emotional wellbeing and clarify that wellbeing does not simply mean always experiencing positive emotions. Introduce the idea that there are different models of emotional intelligence.)  '''Research needed:''' A foundational source defining emotional intelligence, such as Mayer and Salovey, plus a reliable source defining emotional/subjective wellbeing 
Emotional intelligence refers to the ability to recognise, understand, use, and manage emotions (Givon et al., 2020). There are several approaches to understanding EI. Ability models view EI as a form of intelligence, trait models focus on people's perceptions of their own emotional abilities, and mixed models combine emotional abilities with personality characteristics (Luna et al., 2021). One of the most influential ability approaches is Mayer and Salovey's four-branch model. This model describes EI through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions (Salovey &amp; Grewal, 2005). In comparison, trait EI focuses more on how people perceive their own emotional abilities and tendencies. 

Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. It is commonly examined as part of subjective wellbeing, which considers both how people feel and how satisfied they are with their lives (Villanueva et al., 2020). Emotional wellbeing is not simply the absence of mental illness, nor does it mean feeling happy or positive all the time (Park et al., 2022). Negative emotions such as sadness, anger, anxiety, and disappointment are a normal part of life. Instead, emotional wellbeing involves being able to experience and respond to different emotions while maintaining positive emotional functioning and overall life satisfaction.

* '''What is the relationship between emotional intelligence and emotional wellbeing?'''
Research shows that emotional intelligence is linked to greater wellbeing. Xu et al. (2021) found a moderate positive relationship between emotional intelligence and subjective wellbeing (''r'' = .32), meaning people with higher emotional intelligence often report feeling better. A large meta-analysis of over 3,000 studies and more than one million participants also found a positive link between emotional intelligence and human flourishing (''r'' = .28) (Robinson &amp; Zell, 2026). The strength of this connection varied across studies and was stronger when both emotional intelligence and flourishing were measured with self-report tools. Results can also change depending on which model of emotional intelligence is used and how wellbeing is measured, such as through emotional experiences, life satisfaction, or other factors (Llamas-Díaz et al., 2022). In summary, emotional intelligence appears to be related to greater wellbeing, but the strength of this link depends on how both are defined and measured.

* (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

'''Figure 2. The four-branch model of emotional intelligence'''

Create a simple diagram showing:

Perceiving emotions → Using emotions → Understanding emotions → Managing emotions

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===

* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; 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/

Park, C. L., Kubzansky, L. D., Chafouleas, S. M., Davidson, R. J., Keltner, D., Parsafar, P., Conwell, Y., Martin, M. Y., Hanmer, J., &amp; Wang, K. H. (2022). Emotional well-being: What it is and why it matters. Affective Science, 4(1). https://doi.org/10.1007/s42761-022-00163-0

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://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]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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[[Category:Motivation and emotion/Book/Emotional intelligence]]
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      <text bytes="27903" sha1="ctry2d0bdddr18qechnprcls4nuyfje" xml:space="preserve">{{title|Emotional intelligence:&lt;br&gt;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 result.]]

'''Scenario'''

Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.
* '''What are emotional intelligence and emotional wellbeing?''' 
Emotional intelligence refers to the ability to recognise, understand, use, and manage emotions (Givon et al., 2020). There are several approaches to understanding EI. Ability models view EI as a form of intelligence, trait models focus on people's perceptions of their own emotional abilities, and mixed models combine emotional abilities with personality characteristics (Luna et al., 2021). One of the most influential ability approaches is Mayer and Salovey's four-branch model. This model describes EI through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions (Salovey &amp; Grewal, 2005). In comparison, trait EI focuses more on how people perceive their own emotional abilities and tendencies. 

Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. It is commonly examined as part of subjective wellbeing, which considers both how people feel and how satisfied they are with their lives (Villanueva et al., 2020). Emotional wellbeing is not simply the absence of mental illness, nor does it mean feeling happy or positive all the time (Park et al., 2022). Negative emotions such as sadness, anger, anxiety, and disappointment are a normal part of life. Instead, emotional wellbeing involves being able to experience and respond to different emotions while maintaining positive emotional functioning and overall life satisfaction.

* '''What is the relationship between emotional intelligence and emotional wellbeing?'''
Research shows that emotional intelligence is linked to greater wellbeing. Xu et al. (2021) found a moderate positive relationship between emotional intelligence and subjective wellbeing (''r'' = .32), meaning people with higher emotional intelligence often report feeling better. A large meta-analysis of over 3,000 studies and more than one million participants also found a positive link between emotional intelligence and human flourishing (''r'' = .28) (Robinson &amp; Zell, 2026). The strength of this connection varied across studies and was stronger when both emotional intelligence and flourishing were measured with self-report tools. Results can also change depending on which model of emotional intelligence is used and how wellbeing is measured, such as through emotional experiences, life satisfaction, or other factors (Llamas-Díaz et al., 2022). In summary, emotional intelligence appears to be related to greater wellbeing, but the strength of this link depends on how both are defined and measured.

* (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

'''Figure 2. The four-branch model of emotional intelligence'''

Create a simple diagram showing:

Perceiving emotions → Using emotions → Understanding emotions → Managing emotions

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===

* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; 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/

Park, C. L., Kubzansky, L. D., Chafouleas, S. M., Davidson, R. J., Keltner, D., Parsafar, P., Conwell, Y., Martin, M. Y., Hanmer, J., &amp; Wang, K. H. (2022). Emotional well-being: What it is and why it matters. Affective Science, 4(1). https://doi.org/10.1007/s42761-022-00163-0

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://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]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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__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 result.]]

'''Scenario'''

Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.

'''What are emotional intelligence and emotional wellbeing?''' 
Emotional intelligence refers to the ability to recognise, understand, use, and manage emotions (Givon et al., 2020). There are several approaches to understanding EI. Ability models view EI as a form of intelligence, trait models focus on people's perceptions of their own emotional abilities, and mixed models combine emotional abilities with personality characteristics (Luna et al., 2021). One of the most influential ability approaches is Mayer and Salovey's four-branch model. This model describes EI through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions (Salovey &amp; Grewal, 2005). In comparison, trait EI focuses more on how people perceive their own emotional abilities and tendencies. 

Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. It is commonly examined as part of subjective wellbeing, which considers both how people feel and how satisfied they are with their lives (Villanueva et al., 2020). Emotional wellbeing is not simply the absence of mental illness, nor does it mean feeling happy or positive all the time (Park et al., 2022). Negative emotions such as sadness, anger, anxiety, and disappointment are a normal part of life. Instead, emotional wellbeing involves being able to experience and respond to different emotions while maintaining positive emotional functioning and overall life satisfaction.

'''What is the relationship between emotional intelligence and emotional wellbeing?'''
Research shows that emotional intelligence is linked to greater wellbeing. Xu et al. (2021) found a moderate positive relationship between emotional intelligence and subjective wellbeing (''r'' = .32), meaning people with higher emotional intelligence often report feeling better. A large meta-analysis of over 3,000 studies and more than one million participants also found a positive link between emotional intelligence and human flourishing (''r'' = .28) (Robinson &amp; Zell, 2026). The strength of this connection varied across studies and was stronger when both emotional intelligence and flourishing were measured with self-report tools. Results can also change depending on which model of emotional intelligence is used and how wellbeing is measured, such as through emotional experiences, life satisfaction, or other factors (Llamas-Díaz et al., 2022). In summary, emotional intelligence appears to be related to greater wellbeing, but the strength of this link depends on how both are defined and measured.

* (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

'''Figure 2. The four-branch model of emotional intelligence'''

Create a simple diagram showing:

Perceiving emotions → Using emotions → Understanding emotions → Managing emotions

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===

* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; 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/

Park, C. L., Kubzansky, L. D., Chafouleas, S. M., Davidson, R. J., Keltner, D., Parsafar, P., Conwell, Y., Martin, M. Y., Hanmer, J., &amp; Wang, K. H. (2022). Emotional well-being: What it is and why it matters. Affective Science, 4(1). https://doi.org/10.1007/s42761-022-00163-0

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://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]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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__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 result.]]

'''Scenario'''

Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.

'''What is the relationship between emotional intelligence and emotional wellbeing?'''
Research shows that emotional intelligence is linked to greater wellbeing. Xu et al. (2021) found a moderate positive relationship between emotional intelligence and subjective wellbeing (''r'' = .32), meaning people with higher emotional intelligence often report feeling better. A large meta-analysis of over 3,000 studies and more than one million participants also found a positive link between emotional intelligence and human flourishing (''r'' = .28) (Robinson &amp; Zell, 2026). The strength of this connection varied across studies and was stronger when both emotional intelligence and flourishing were measured with self-report tools. Results can also change depending on which model of emotional intelligence is used and how wellbeing is measured, such as through emotional experiences, life satisfaction, or other factors (Llamas-Díaz et al., 2022). In summary, emotional intelligence appears to be related to greater wellbeing, but the strength of this link depends on how both are defined and measured. 

* (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
Emotional intelligence refers to the ability to recognise, understand, use, and manage emotions (Givon et al., 2020). There are several approaches to understanding EI. Ability models view EI as a form of intelligence, trait models focus on people's perceptions of their own emotional abilities, and mixed models combine emotional abilities with personality characteristics (Luna et al., 2021). One of the most influential ability approaches is Mayer and Salovey's four-branch model. This model describes EI through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions (Salovey &amp; Grewal, 2005). In comparison, trait EI focuses more on how people perceive their own emotional abilities and tendencies.

Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. It is commonly examined as part of subjective wellbeing, which considers both how people feel and how satisfied they are with their lives (Villanueva et al., 2020). Emotional wellbeing is not simply the absence of mental illness, nor does it mean feeling happy or positive all the time (Park et al., 2022). Negative emotions such as sadness, anger, anxiety, and disappointment are a normal part of life. Instead, emotional wellbeing involves being able to experience and respond to different emotions while maintaining positive emotional functioning and overall life satisfaction.
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

'''Figure 2. The four-branch model of emotional intelligence'''

Create a simple diagram showing:

Perceiving emotions → Using emotions → Understanding emotions → Managing emotions

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

Research shows that emotional intelligence is linked to greater wellbeing. Xu et al. (2021) found a moderate positive relationship between emotional intelligence and subjective wellbeing (''r'' = .32), meaning people with higher emotional intelligence often report feeling better. A large meta-analysis of over 3,000 studies and more than one million participants also found a positive link between emotional intelligence and human flourishing (''r'' = .28) (Robinson &amp; Zell, 2026). The strength of this connection varied across studies and was stronger when both emotional intelligence and flourishing were measured with self-report tools. Results can also change depending on which model of emotional intelligence is used and how wellbeing is measured, such as through emotional experiences, life satisfaction, or other factors (Llamas-Díaz et al., 2022). In summary, emotional intelligence appears to be related to greater wellbeing, but the strength of this link depends on how both are defined and measured.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===

* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; 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/

Park, C. L., Kubzansky, L. D., Chafouleas, S. M., Davidson, R. J., Keltner, D., Parsafar, P., Conwell, Y., Martin, M. Y., Hanmer, J., &amp; Wang, K. H. (2022). Emotional well-being: What it is and why it matters. Affective Science, 4(1). https://doi.org/10.1007/s42761-022-00163-0

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://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]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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__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 result.]]

'''Scenario'''

Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.

'''What is the relationship between emotional intelligence and emotional wellbeing?'''
Research shows that emotional intelligence is linked to greater wellbeing. Xu et al. (2021) found a moderate positive relationship between emotional intelligence and subjective wellbeing (''r'' = .32), meaning people with higher emotional intelligence often report feeling better. A large meta-analysis of over 3,000 studies and more than one million participants also found a positive link between emotional intelligence and human flourishing (''r'' = .28) (Robinson &amp; Zell, 2026). The strength of this connection varied across studies and was stronger when both emotional intelligence and flourishing were measured with self-report tools. Results can also change depending on which model of emotional intelligence is used and how wellbeing is measured, such as through emotional experiences, life satisfaction, or other factors (Llamas-Díaz et al., 2022). In summary, emotional intelligence appears to be related to greater wellbeing, but the strength of this link depends on how both are defined and measured. 

* (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
Emotional intelligence refers to the ability to recognise, understand, use, and manage emotions (Givon et al., 2020). There are several approaches to understanding EI. Ability models view EI as a form of intelligence, trait models focus on people's perceptions of their own emotional abilities, and mixed models combine emotional abilities with personality characteristics (Luna et al., 2021). One of the most influential ability approaches is Mayer and Salovey's four-branch model. This model describes EI through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions (Salovey &amp; Grewal, 2005). In comparison, trait EI focuses more on how people perceive their own emotional abilities and tendencies.

Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. It is commonly examined as part of subjective wellbeing, which considers both how people feel and how satisfied they are with their lives (Villanueva et al., 2020). Emotional wellbeing is not simply the absence of mental illness, nor does it mean feeling happy or positive all the time (Park et al., 2022). Negative emotions such as sadness, anger, anxiety, and disappointment are a normal part of life. Instead, emotional wellbeing involves being able to experience and respond to different emotions while maintaining positive emotional functioning and overall life satisfaction.
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

Research shows that emotional intelligence is linked to greater wellbeing. Xu et al. (2021) found a moderate positive relationship between emotional intelligence and subjective wellbeing (''r'' = .32), meaning people with higher emotional intelligence often report feeling better. A large meta-analysis of over 3,000 studies and more than one million participants also found a positive link between emotional intelligence and human flourishing (''r'' = .28) (Robinson &amp; Zell, 2026). The strength of this connection varied across studies and was stronger when both emotional intelligence and flourishing were measured with self-report tools. Results can also change depending on which model of emotional intelligence is used and how wellbeing is measured, such as through emotional experiences, life satisfaction, or other factors (Llamas-Díaz et al., 2022). In summary, emotional intelligence appears to be related to greater wellbeing, but the strength of this link depends on how both are defined and measured.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===

* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; 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/

Park, C. L., Kubzansky, L. D., Chafouleas, S. M., Davidson, R. J., Keltner, D., Parsafar, P., Conwell, Y., Martin, M. Y., Hanmer, J., &amp; Wang, K. H. (2022). Emotional well-being: What it is and why it matters. Affective Science, 4(1). https://doi.org/10.1007/s42761-022-00163-0

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://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]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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__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 result.]]

'''Scenario'''

Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.

'''What is the relationship between emotional intelligence and emotional wellbeing?''' 

* (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
Emotional intelligence refers to the ability to recognise, understand, use, and manage emotions (Givon et al., 2020). There are several approaches to understanding EI. Ability models view EI as a form of intelligence, trait models focus on people's perceptions of their own emotional abilities, and mixed models combine emotional abilities with personality characteristics (Luna et al., 2021). One of the most influential ability approaches is Mayer and Salovey's four-branch model. This model describes EI through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions (Salovey &amp; Grewal, 2005). In comparison, trait EI focuses more on how people perceive their own emotional abilities and tendencies.

Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. It is commonly examined as part of subjective wellbeing, which considers both how people feel and how satisfied they are with their lives (Villanueva et al., 2020). Emotional wellbeing is not simply the absence of mental illness, nor does it mean feeling happy or positive all the time (Park et al., 2022). Negative emotions such as sadness, anger, anxiety, and disappointment are a normal part of life. Instead, emotional wellbeing involves being able to experience and respond to different emotions while maintaining positive emotional functioning and overall life satisfaction.
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

Research shows that emotional intelligence is linked to greater wellbeing. Xu et al. (2021) found a moderate positive relationship between emotional intelligence and subjective wellbeing (''r'' = .32), meaning people with higher emotional intelligence often report feeling better. A large meta-analysis of over 3,000 studies and more than one million participants also found a positive link between emotional intelligence and human flourishing (''r'' = .28) (Robinson &amp; Zell, 2026). The strength of this connection varied across studies and was stronger when both emotional intelligence and flourishing were measured with self-report tools. Results can also change depending on which model of emotional intelligence is used and how wellbeing is measured, such as through emotional experiences, life satisfaction, or other factors (Llamas-Díaz et al., 2022). In summary, emotional intelligence appears to be related to greater wellbeing, but the strength of this link depends on how both are defined and measured.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===

* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; 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/

Park, C. L., Kubzansky, L. D., Chafouleas, S. M., Davidson, R. J., Keltner, D., Parsafar, P., Conwell, Y., Martin, M. Y., Hanmer, J., &amp; Wang, K. H. (2022). Emotional well-being: What it is and why it matters. Affective Science, 4(1). https://doi.org/10.1007/s42761-022-00163-0

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

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** Incorrect capitalisation
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** Citing sources that haven't been consulted
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==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://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]

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[[Category:Motivation and emotion/Book/Emotional intelligence]]
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== 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 result.]]

'''Scenario'''

Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.

'''What is the relationship between emotional intelligence and emotional wellbeing?''' 

* (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
Emotional intelligence refers to the ability to recognise, understand, use, and manage emotions (Givon et al., 2020). There are several approaches to understanding EI. Ability models view EI as a form of intelligence, trait models focus on people's perceptions of their own emotional abilities, and mixed models combine emotional abilities with personality characteristics (Luna et al., 2021). One of the most influential ability approaches is Mayer and Salovey's four-branch model. This model describes EI through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions (Salovey &amp; Grewal, 2005). In comparison, trait EI focuses more on how people perceive their own emotional abilities and tendencies.

Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. It is commonly examined as part of subjective wellbeing, which considers both how people feel and how satisfied they are with their lives (Villanueva et al., 2020). Emotional wellbeing is not simply the absence of mental illness, nor does it mean feeling happy or positive all the time (Park et al., 2022). Negative emotions such as sadness, anger, anxiety, and disappointment are a normal part of life. Instead, emotional wellbeing involves being able to experience and respond to different emotions while maintaining positive emotional functioning and overall life satisfaction.
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

Research shows that emotional intelligence is linked to greater wellbeing. Xu et al. (2021) found a moderate positive relationship between emotional intelligence and subjective wellbeing (''r'' = .32), meaning people with higher emotional intelligence often report feeling better. A large meta-analysis of over 3,000 studies and more than one million participants also found a positive link between emotional intelligence and human flourishing (''r'' = .28) (Robinson &amp; Zell, 2026). The strength of this connection varied across studies and was stronger when both emotional intelligence and flourishing were measured with self-report tools. Results can also change depending on which model of emotional intelligence is used and how wellbeing is measured, such as through emotional experiences, life satisfaction, or other factors (Llamas-Díaz et al., 2022). In summary, emotional intelligence appears to be related to greater wellbeing, but the strength of this link depends on how both are defined and measured.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===

* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== How can emotional intelligence be developed to support emotional wellbeing? ==
Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; 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/

Park, C. L., Kubzansky, L. D., Chafouleas, S. M., Davidson, R. J., Keltner, D., Parsafar, P., Conwell, Y., Martin, M. Y., Hanmer, J., &amp; Wang, K. H. (2022). Emotional well-being: What it is and why it matters. Affective Science, 4(1). https://doi.org/10.1007/s42761-022-00163-0

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://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]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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[[Category:Motivation and emotion/Book/Emotional intelligence]]
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== 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 who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.

'''What is the relationship between emotional intelligence and emotional wellbeing?''' 

* (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
Emotional intelligence refers to the ability to recognise, understand, use, and manage emotions (Givon et al., 2020). There are several approaches to understanding EI. Ability models view EI as a form of intelligence, trait models focus on people's perceptions of their own emotional abilities, and mixed models combine emotional abilities with personality characteristics (Luna et al., 2021). One of the most influential ability approaches is Mayer and Salovey's four-branch model. This model describes EI through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions (Salovey &amp; Grewal, 2005). In comparison, trait EI focuses more on how people perceive their own emotional abilities and tendencies.

Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. It is commonly examined as part of subjective wellbeing, which considers both how people feel and how satisfied they are with their lives (Villanueva et al., 2020). Emotional wellbeing is not simply the absence of mental illness, nor does it mean feeling happy or positive all the time (Park et al., 2022). Negative emotions such as sadness, anger, anxiety, and disappointment are a normal part of life. Instead, emotional wellbeing involves being able to experience and respond to different emotions while maintaining positive emotional functioning and overall life satisfaction.
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

Research shows that emotional intelligence is linked to greater wellbeing. Xu et al. (2021) found a moderate positive relationship between emotional intelligence and subjective wellbeing (''r'' = .32), meaning people with higher emotional intelligence often report feeling better. A large meta-analysis of over 3,000 studies and more than one million participants also found a positive link between emotional intelligence and human flourishing (''r'' = .28) (Robinson &amp; Zell, 2026). The strength of this connection varied across studies and was stronger when both emotional intelligence and flourishing were measured with self-report tools. Results can also change depending on which model of emotional intelligence is used and how wellbeing is measured, such as through emotional experiences, life satisfaction, or other factors (Llamas-Díaz et al., 2022). In summary, emotional intelligence appears to be related to greater wellbeing, but the strength of this link depends on how both are defined and measured.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===

* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== How can emotional intelligence be developed to support emotional wellbeing? ==
Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; 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/

Park, C. L., Kubzansky, L. D., Chafouleas, S. M., Davidson, R. J., Keltner, D., Parsafar, P., Conwell, Y., Martin, M. Y., Hanmer, J., &amp; Wang, K. H. (2022). Emotional well-being: What it is and why it matters. Affective Science, 4(1). https://doi.org/10.1007/s42761-022-00163-0

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://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]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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__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 who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.

'''What is the relationship between emotional intelligence and emotional wellbeing?''' 

* (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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?
# How can emotional intelligence be developed to support emotional wellbeing?
# Which components of emotional intelligence are particularly important for emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
Emotional intelligence refers to the ability to recognise, understand, use, and manage emotions (Givon et al., 2020). There are several approaches to understanding EI. Ability models view EI as a form of intelligence, trait models focus on people's perceptions of their own emotional abilities, and mixed models combine emotional abilities with personality characteristics (Luna et al., 2021). One of the most influential ability approaches is Mayer and Salovey's four-branch model. This model describes EI through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions (Salovey &amp; Grewal, 2005). In comparison, trait EI focuses more on how people perceive their own emotional abilities and tendencies.

Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. It is commonly examined as part of subjective wellbeing, which considers both how people feel and how satisfied they are with their lives (Villanueva et al., 2020). Emotional wellbeing is not simply the absence of mental illness, nor does it mean feeling happy or positive all the time (Park et al., 2022). Negative emotions such as sadness, anger, anxiety, and disappointment are a normal part of life. Instead, emotional wellbeing involves being able to experience and respond to different emotions while maintaining positive emotional functioning and overall life satisfaction.
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===
Research shows that emotional intelligence is linked to greater wellbeing. Xu et al. (2021) found a moderate positive relationship between emotional intelligence and subjective wellbeing (''r'' = .32), meaning people with higher emotional intelligence often report feeling better. A large meta-analysis of over 3,000 studies and more than one million participants also found a positive link between emotional intelligence and human flourishing (''r'' = .28) (Robinson &amp; Zell, 2026). The strength of this connection varied across studies and was stronger when both emotional intelligence and flourishing were measured with self-report tools. Results can also change depending on which model of emotional intelligence is used and how wellbeing is measured, such as through emotional experiences, life satisfaction, or other factors (Llamas-Díaz et al., 2022). In summary, emotional intelligence appears to be related to greater wellbeing, but the strength of this link depends on how both are defined and measured.
* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== How can emotional intelligence be developed to support emotional wellbeing? ==
Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; 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/

Park, C. L., Kubzansky, L. D., Chafouleas, S. M., Davidson, R. J., Keltner, D., Parsafar, P., Conwell, Y., Martin, M. Y., Hanmer, J., &amp; Wang, K. H. (2022). Emotional well-being: What it is and why it matters. Affective Science, 4(1). https://doi.org/10.1007/s42761-022-00163-0

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://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]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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__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 who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.

'''What is the relationship between emotional intelligence and emotional wellbeing?''' 

* (Discuss research showing whether higher emotional intelligence is associated with greater emotional or psychological wellbeing. Consider outcomes such as positive affect, life satisfaction, happiness, stress, and negative emotional experiences. Be careful to distinguish an association from evidence that emotional intelligence directly causes better wellbeing.)  '''Research needed:''' Meta-analyses or empirical studies examining the association between emotional intelligence and wellbeing (Robinson &amp; Zell, 2026)
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

* {{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? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
Emotional intelligence refers to the ability to recognise, understand, use, and manage emotions (Givon et al., 2020). There are several approaches to understanding EI. Ability models view EI as a form of intelligence, trait models focus on people's perceptions of their own emotional abilities, and mixed models combine emotional abilities with personality characteristics (Luna et al., 2021). One of the most influential ability approaches is Mayer and Salovey's four-branch model. This model describes EI through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions (Salovey &amp; Grewal, 2005). In comparison, trait EI focuses more on how people perceive their own emotional abilities and tendencies.

Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. It is commonly examined as part of subjective wellbeing, which considers both how people feel and how satisfied they are with their lives (Villanueva et al., 2020). Emotional wellbeing is not simply the absence of mental illness, nor does it mean feeling happy or positive all the time (Park et al., 2022). Negative emotions such as sadness, anger, anxiety, and disappointment are a normal part of life. Instead, emotional wellbeing involves being able to experience and respond to different emotions while maintaining positive emotional functioning and overall life satisfaction.
* Introduce emotional intelligence.
* Explain that EI concerns the processing and management of emotional information.
* Explain that EI is not simply &quot;being emotional&quot; or &quot;being a nice person&quot;.
* Introduce major conceptualisations of EI.
* Explain why differences in definitions matter when examining research.

==== '''Ability model of emotional intelligence''' ====
Introduce Mayer and Salovey's ability model.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

&lt;quiz display=simple&gt;
{Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ 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.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===
Research shows that emotional intelligence is linked to greater wellbeing. Xu et al. (2021) found a moderate positive relationship between emotional intelligence and subjective wellbeing (''r'' = .32), meaning people with higher emotional intelligence often report feeling better. A large meta-analysis of over 3,000 studies and more than one million participants also found a positive link between emotional intelligence and human flourishing (''r'' = .28) (Robinson &amp; Zell, 2026). The strength of this connection varied across studies and was stronger when both emotional intelligence and flourishing were measured with self-report tools. Results can also change depending on which model of emotional intelligence is used and how wellbeing is measured, such as through emotional experiences, life satisfaction, or other factors (Llamas-Díaz et al., 2022). In summary, emotional intelligence appears to be related to greater wellbeing, but the strength of this link depends on how both are defined and measured.
* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''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, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together 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 concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 1.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== How can emotional intelligence be developed to support emotional wellbeing? ==
Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* 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==
;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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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

;
;
'''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]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her 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==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[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)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* 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:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; 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 

Chen, C., Kotozaki , Y., Okubo , R., &amp; 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/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; 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/

Park, C. L., Kubzansky, L. D., Chafouleas, S. M., Davidson, R. J., Keltner, D., Parsafar, P., Conwell, Y., Martin, M. Y., Hanmer, J., &amp; Wang, K. H. (2022). Emotional well-being: What it is and why it matters. Affective Science, 4(1). https://doi.org/10.1007/s42761-022-00163-0

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://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]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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&lt;div align=center&gt;[[Motivation and emotion/Book/2025|2026 list of topics]].&lt;/div&gt;
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:Stockcouple.jpg|thumb|'''Figure 1'''. Stock image of couple hugging]]

; Picture this...

Jim and Pam have been dating for a few months; things have been going well but at times there is noticeable emotional distance between them. 

One night Jim tells Pam that “I haven’t talked to anyone about this, but I’m terrified one day you will get bored and leave”, this confession opens the flood gates for Jim and Pam to share their deepest fears and anxieties. 

In this moment, an invisible barrier dissolves, they have discovered a new profound sense of trust and understanding. A new and deeper layer of their relationship has been unlocked; there is an emotional closeness that wasn’t there before.

Why does this self-disclosure from Jim create this emotional closeness between him and Pam?
{{RoundBoxBottom}}

Key points
* What is self-disclosure?
* Brief overview of some self-disclosure theories (?)
* Brief overview of the importance of emotional closeness

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What is the relationship between self-disclosure and emotional intimacy?
* What role does reciprocal self-disclosure play in emotional closeness in relationships?
* How can self-disclosure in relationships be fostered?
{{RoundBoxBottom}}

The Overview section (180 to 330 words) (ignore)
# '''Scenario''': A short, engaging case study or real-world example in a feature box, with an accompanying image (see above)
# '''Explanation of the problem, issue, or topic''': Briefly explain the problem, why it is important, and outline how psychological science can help
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box
Resources (ignore):
* [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] explains about how to edit
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
[[File:Bdspt.model.webp|'''Figure 2.''' Social Penetration Theory (1973) Model.|alt=Figure 2. Social Penetration Theory (1973) Model.|thumb]]

==What is Self-Disclosure?==

* Define in [[wikipedia:Self-disclosure|self-disclosure]] in further depth
* Different types of self-disclosure
* Self-disclosure theories (subheadings): 
** Social Penetration Theory (Altman &amp; Taylor, 1973) (See figure 2),
** Social Exchange Theory (Thibaut &amp; Kelley, 1959) rewards vs costs of self disclosure

;Quiz
&lt;quiz display=simple&gt;
{Who developed social penetration theory?
|type=&quot;()&quot;}
+ Irwin Altman &amp; Dalmas Taylor
- Carl Rogers &amp; Abraham Maslow
&lt;/quiz&gt;

== Self-Disclosure &amp; Emotional Intimacy ==

*What is [[wikipedia:Emotional_intimacy|emotional intimacy]]? (definition)
*Why we strive for emotional intimacy
*Different types of relationships &amp; emotional intimacy - stranger on the train phenomenon
*How can self-disclosure increase emotional intimacy? How does emotional intimacy build relationships?

==What is Reciprocal Self-Disclosure?==

* Definition
* Norm of reciprocity
* The Dyadic Effect
* Breadth &amp; Depth (SPT, 1973)
* Building trust&lt;br /&gt;

==Promoting Self-Disclosure==
* How to encourage self-disclosure
* How to respond to self-disclosures
* Boundaries

=== Learning features (ignore) ===
{{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., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{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=&quot;wikitable&quot; style=&quot;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
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* How does self-disclosure foster emotional closeness in relationships? Signals vulnerability, builds trust, increases understanding, and progresses relationships.
* What is the relationship between self-disclosure and emotional intimacy? Self-disclosure drives emotional intimacy...
* What role does reciprocal self-disclosure play in emotional closeness in relationships? Shared cycle of vulnerability, trust, mutual understanding.
* How can self-disclosure in relationships be fostered? Start slow, reciprocity! Validation &amp; listening. Match pacing and comfortability.
* Take home messages

Notes (ignore):
*The Conclusion is arguably the most important section (150-330 words)
* 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

{{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/2026/Responsiveness and interpersonal trust|Responsiveness and interpersonal trust]] (Book chapter, 2026)
* [[Motivation and emotion/Book/2025/Self-disclosure and well-being|Self-disclosure and well-being]] (Book chapter, 2025)
* [[wikipedia:Social_exchange_theory|Social exchange theory]] (Wikipedia)
* [[wikipedia:Social_penetration_theory|Social penetration theory]] (Wikipedia)

==References==
{{Hanging indent|1=

Altman, I., &amp; Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Halsted Press. https://psycnet.apa.org/record/1973-28661-000

Derlega, V. J., Winstead, B. A., &amp; Greene, K. (2008). Self-disclosure and starting a close relationship. In S. Sprecher, A. Wenzel, &amp; J. Harvey (Eds.), ''Handbook of relationship initiation''. Psychology Press. 53–174. https://psycnet.apa.org/record/2008-09972-008

Forgas, J. P. (2011). Affective influences on self-disclosure: Mood effects on the intimacy and reciprocity of disclosing personal information. ''Journal of Personality and Social Psychology, 100''(3). 449-461. https://doi.org/10.1037/a0021129

Greene, K., Derlega, V. J., &amp; Mathews, A. (2006). Self-disclosure in personal relationships. In A. L. Vangelisti &amp; D. Perlman (Eds.), ''The Cambridge handbook of personal relationships.'' Cambridge University Press. 409-427. https://doi.org/10.1017/CBO9780511606632.023 

Pecune, F. (2013). Toward a computational model of social relations for artificial companions. ''Proceedings - 2013 Humaine Association Conference on Affective Computing and Intelligent Interaction''. 677-682. [https://www.researchgate.net/publication/261271664%20Toward%20a%20Computational%20Model%20of%20Social%20Relations%20for%20Artificial%20Companions DOI:10.1109/ACII.2013.118]

Tolstedt, B. E., &amp; Stokes, J. P. (1984). Self-disclosure, intimacy, and the depenetration process. ''Journal of Personality and Social Psychology, 46''(1). 84-90. https://doi.org/10.1037/0022-3514.46.1.84

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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. 

* [https://www.youtube.com/watch?v=Z4bRXozREKU&amp;t=90s Self-disclosure in personal relationships: How to stop oversharing] (youtube.com)
* [https://www.psychologytoday.com/au/blog/between-the-sheets/202011/why-romantic-intimacy-requires-self-disclosure Why romantic intimacy requires self-disclosure] (psychologytoday.com)

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__TOC__

==Overview==
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&lt;div style=&quot;border: 2px solid #a2a9b1; border-radius: 15px; padding: 12px; background-color: #f8f9fa; width: 150px; float: right; margin: 0.6em 0 10px 10px;&quot;&gt;
[[File:Student rushing to school.png|center|none|150px]]
&lt;div style=&quot;font-size: 85%; color: #333333; line-height: 1.4; padding-top: 8px; text-align: left;&quot;&gt;
'''Figure 1'''. A student rushing to school (AI-Generated).
&lt;/div&gt;
&lt;/div&gt;
Picture this: Your alarm rings at 8:30am, just in time to start school at 9:00. You roll out of bed, get dressed, brush your teeth, and grab a snack before heading out the door to leave for school. The walk is only 5 minutes away, and by this time, it is 8:50. With just 5 minutes to spare, you arrive at school and walk into class. You begin to chat with Jane, a fellow classmate and neighbour, who tells you that she arrived at school at 8:00am; well before you were even awake. You sit there while she talks about her morning, thinking “Why can I never make it to school as early as Jane does?”. You and Jane live next door to each other, are in the same grade, and attend the same classes. You are only ever on time when school finishes, leaving right as the final bell rings, while Jane often stays back after class. So, why is your daily routine so different from Jane’s? 

{{RoundBoxBottom}}

'''Values''' are at the forefront of many social and personal situations, acting as moral compasses guiding how we behave and interact with others. In this chapter, we look at how an alignment between '''personal and situational values''' can influence '''motivation''' throughout our day-to-day lives. The following dot points briefly explain the problem, why it is important, and outlines how psychological science can help;

* When personal and situational values are misaligned, this can create value incongruence.
* Value incongruence is important to generate autonomous, and controlled motivation.
* Consequences of value incongruence include burnout (e.g., job), lowered intrinsic motivation, and reduced engagement in daily tasks or activities.
* Motivational science purports several notable psychological theories as both solutions and explanations for why value congruence or incongruence occurs.
** Such theories include; Self-Determinism Theory (SDT), Self-Concordance Theory, Motivational Congruence Theory, Person-Organisation (P-O) Fit, and Amity Goal Orientation Theory.

{{RoundBoxTop|theme=3}}
📍 '''Focus questions'''

* What are personal and situational values?
* How do personal and situational values become aligned or misaligned?
* How does value incongruence influence motivation?
* When does value congruence strengthen or weaken motivation?
* How can understanding value congruence help support motivation?
{{RoundBoxBottom}}

==What are personal and situational values?==
Introductory paragraph: Will discuss what values are and how they are relevant to human behaviour and motivation. Will then state that values can be differentiated into personal and situational values...

=== '''Understanding Personal Values''' ===
* &lt;u&gt;Schwartz (2012):&lt;/u&gt;
** Theoretical underpinnings for understanding personal values, defining them as beliefs about desirable goals that influence people's decision-making and behaviour.
** Introduces different types of values, including self-direction, security, benevolence, and achievement.
** ''*Authors note: Will fill out a table with definitions of different value types below*'' 
*&lt;u&gt;Ryan &amp; Deci (2018):&lt;/u&gt;
**Discusses how personally held values relate to self-endorsed behaviour and autonomous motivation. 
**Explain why pursuing something deemed 'meaningful' and reflecting one's own values can feel more enriching.  
{| class=&quot;wikitable&quot;
|+''*Table 1.'' Types of Personal Values
! colspan=&quot;2&quot; |Schwartz (2012) proposes 10 basic personal values:
|-
|1. Self-Direction
|...
|-
|2. Stimulation
|...
|-
|3. Hedonism
|...
|-
|4. Achievement
|...
|-
|5. Power
|...
|-
|6. Security
|...
|-
|7. Conformity
|...
|-
|8. Tradition
|...
|-
|9. Benevolence
|...
|-
|10. Universalism
|...                                              
|}

=== '''Understanding Situational Values''' ===
* &lt;u&gt;van Vianen (2018):&lt;/u&gt;
** Discusses person-environment fit, conveying an understanding of how characteristics of an environment can be compatible or incompatible with individuals characteristics and values.
** Often applied in organisational contexts - this will be discussed further in detail below.
** Conceptualises situational values as values communicated, encouraged, or prioritised within an individual's environment.
*&lt;u&gt;Kristof (1996):&lt;/u&gt;
**A seminal article viewing organisational values as part of the environment in which people operate.
**States that compatibility may occur when an individual and organisation share similar core characteristics or values.
=== '''Values and Goal Pursuit''' ===
* &lt;u&gt;Schartwz (2012):&lt;/u&gt;
** Describes values as guiding principles that influence goal selection and behaviour - specifically the behaviour related to pursuing goals.
** Argues that goals are more likely to be pursued when they align with personal values.
*&lt;u&gt;Ryan &amp; Deci (2018):&lt;/u&gt;
**States that goals and behaviours may be pursued for different reasons, with a focus on autonomous motivation - when behaviour is personally endorsed and consistent with values.
**Bridges the gap between what people value, what goals they pursue, and why they pursue them.


== How do personal and situational values become aligned? ==
Introductory paragraph: Will discuss the fundamentals of how personal and situational values become aligned. Will introduce motivation theories...

=== '''Understanding Value Congruence''' ===
* &lt;u&gt;Kristof (1996):&lt;/u&gt;
** Provides a basis for understanding value congruence, particularly that when 'fit' occurs when individuals and organisations share compatible values.
** Congruence is discussed as a 'match' between individual values and environmental values/provisions.
*&lt;u&gt;van Vianen (2018):&lt;/u&gt;
**Expands on Kristof's (1996) discussion of value congruence, adding that compatibility may occur across multiple contexts rather than being limited to organisations.
**Continues discussion on value congruence through person-environment fit; compatibility occurring between individuals and different aspects of their environment (i.e., some aspects may compatible, some may not be).

=== '''Understanding Value Incongruence''' ===
* &lt;u&gt;Bao et al. (2013):&lt;/u&gt;
** Study providing empirical evidence of value incongruence and its relationship with individual and organisational wellbeing among Catalan nurses.
** An applied example/case study of what happens when individual personal values differ from situational values encouraged in their workplace.
*&lt;u&gt;Hossli et al. (2025):&lt;/u&gt;
**Discusses personal-work value incongruence, proposing that a mismatch between personal and work (i.e., environmental) values produced poorer job-related outcomes.
**Demonstrates that incongruence is far more than having different values, it considers degree of mismatch and extends to consequences for employees. 
[[File:Person-Organisation Fit Diagram.jpg|420x420px|alt=Figure 2. Person-Organisation Fit Scale|thumb|'''''Figure 2.''''' Person-Organisation Fit Scale]]

=== '''Person-Organisation Fit''' ===
*Kristof (1996):
** A key seminal source (briefly mentioned in above sections) for person-organisation fit.
** Compatibility between individuals and organisations can occurs through value similarity - a core concept for understanding person-organisation value congruence.
*&lt;u&gt;Kristof-Brown et al. (2005):&lt;/u&gt;
**Later works of Kristof (1996), a meta-analytic study concerning person-organisation fit, person-job fit, person-group fit, and person-supervisor fit.
**Demonstrates that fit it not as simple as employee-organisation, it extends to various people and aspects of that organisation/environment.
**Emphasises the importance of value congruence for employee/individual outcomes.
*See Figure 2 for a visual representation of the person-organisation fit scale.


== How does value congruence influence motivation? ==
Introductory paragraph: Explains that value congruence can influence motivation - people more likely to invest in goals/activity's that are personally meaningful. Will then use theories to substantiate why this occurs.

=== '''Self-Determination Theory''' ===
* &lt;u&gt;Ryan &amp; Deci (2018):&lt;/u&gt;
** The self-determination theory (SDT) explains that motivation is more autonomous when behaviour is experienced through self-endorsed and consistent values.
** Thus, value congruence supports motivation by allowing individuals to experience their goals/actions as personally meaningful rather then forced upon externally.
*&lt;u&gt;Li et al. (2015):&lt;/u&gt;
**Empirical support for the relationship between value congruence and work engagement - based on autonomous and controlled motivation.
**Connects the principles of SDT to actual workplace motivation - important for applying value congruence in workplace contexts.

=== '''Self-Concordance Theory''' ===
* &lt;u&gt;Ryan &amp; Deci (2018):&lt;/u&gt;
** Supports the idea that goals are more motivating when consistent with individual interests and values.
** Explains why value-aligned goals are more likely to be pursued enthusiastically and continuously. 
* &lt;u&gt;Rawolle et al. (2016):&lt;/u&gt;
**Examines motive incongruence and its relationship with intrinsic motivation.
**Explains what may happen when the motives underlying an individuals behaviour are misaligned with what they want or value in actuality.

=== '''Motivational Congruence Theory''' ===
*&lt;u&gt;Rawolle et al. (2016):&lt;/u&gt;
**Directly examines motive incongruence as well as its relationship with intrinsic motivation and burnout.
**Explains why discrepancies between what people actually want or motivate themselves towards with what they actually do - these behaviours may undermine motivation.
*&lt;u&gt;Hossli et al. (2025):&lt;/u&gt;
**Shows how value incongruence may affect outcomes through multiple psychological processes.
**Demonstrates the complexity of how incongruence translates into reduced positive work outcomes.

=== '''Amity Goal Orientation Theory''' ===
*Schwartz (2012):
**While Schwartz (2012) does not directly test the amity goal orientation theory, it lays the groundwork for this theory in terms of value types (e.g., benevolence, self-directed etc.).
*&lt;u&gt;Levontin &amp; Bardi (2019):&lt;/u&gt;
**Proposes that values can be used to understand the motivational basis of amity goal orientation; a prosocial goal orientation within achievement situations.
**Employs Schwartz's (2012) theory of basic values in a small-scale meta-analysis, findings that the prosocial value of benevolence is positively related to amity goal orientation.
**Power values were found as positively related to performance approach goal orientation, while self-direction values are positively tied to mastery goal orientation, and security values are positively linked to performance-avoidance goal orientation.


== When can value congruence strengthen or weaken motivation? ==
Introductory paragraph: Builds on theoretical mechanisms and discusses value congruence as a complex phenomenon --&gt; not automatically beneficial in every situation.

=== '''When Value Alignment Strengthens Motivation''' ===
* &lt;u&gt;Li et al. (2015):&lt;/u&gt;
** Show that value congruence is positively associated with work engagement, partly through autonomous motivation.
** Argue that alignment is particularly motivating when individuals engage in activities for reasons that they personally endorse.
*&lt;u&gt;Wang et al. (2018):&lt;/u&gt;
**Discuss the operation of value congruence in a transformational leadership and employee behaviour context.
**Explains that alignment may strengthen motivation when organisational/leadership values are communicated in ways employees identify with.
*&lt;u&gt;Lee et al. (2025):&lt;/u&gt;
**Emphasises the importance of leader-follower value congruence - alignment is particularly motivating when employees experience value compatibility with influential people/people in power (e.g., supervisors).

=== '''When Value Incongruence Undermines Motivation''' ===
* &lt;u&gt;Rawolle et al. (2016):&lt;/u&gt;
** Demonstrates how motive incongruence is linked to burnout, with intrinsic motivation playing a mediating role.
** Supports the claim that persistent discrepancies between personal values and pursued goals can undermine motivation and wellbeing.
*&lt;u&gt;Hossli et al. (2025):&lt;/u&gt;
**Examines the effects of personal-work value incongruence on job satisfaction and engagement.
**Demonstrates how misalignment can produce negative outcomes when individuals cannot reconcile their personal values with their work environment. 
*&lt;u&gt;Bao et al. (2013):&lt;/u&gt;
**Discusses the consequences of value misalignment in an applied example (i.e., workplace context).

=== '''Individual Differences in Values and Motivation''' ===
* &lt;u&gt;Schwartz (2012):&lt;/u&gt;
** Individuals differ in the importance they assign to different values.
** Provides an important explanation for why individuals in the same environment may be highly motived and others may be less motivated.
*&lt;u&gt;Ryan &amp; Deci (2018):&lt;/u&gt;
**Describes individuals differences in regard to the extent people experience motivation as autonomous versus controlled.
**Argues that value congruence will not affect everyone in the same way - this describes the inherent nature of individual differences.
*&lt;u&gt;Kristof-Brown et al. (2005):&lt;/u&gt;
**Different types of 'fit' have different relationships with workplace outcomes, which supports the idea that congruence effects depend on which specific aspect of the environment is being evaluated.
* *''Authors note: Will use an example Borderline Personality Disorder (BPD) in regard to individual differences''.*&lt;br /&gt;


== How can understanding value congruence help support motivation? ==
Introductory paragraph: Moves away from theory and research into application. Explains why understanding the relationship between values and motivation is beneficial (e.g., for organisations, goal pursuit). How organisations can support sustained motivation.

=== '''Supporting Value-Aligned Motivation''' ===
* &lt;u&gt;Ryan &amp; Deci (2018):&lt;/u&gt;
** A strong theoretical basis for supporting motivation through autonomy, competence, and relatedness.
** Environments can support motivation by allowing people to understand and personally endorse the purpose behind behaviour. 
*&lt;u&gt;Li et al. (2015):&lt;/u&gt;
**Value congruence can support work engagement and autonomous motivation.
**Value-aligned motivation can be supported by designing roles and goals that allow individuals to connect with their work in a personal and meaningful way.
*&lt;u&gt;Rawolle et al. (2016):&lt;/u&gt;
**Discusses the potential consequences of motive incongruence (e.g., burnout).
**By reducing discrepancies between employee's motives and their work, this can safeguard motivation and support wellbeing.

=== '''Organisational Value Congruence''' ===
* &lt;u&gt;Kristof (1996):&lt;/u&gt;
** Theoretical underpinnings for person-organisation fit to understand and support individuals (e.g., employees) in a workplace setting.
** Value compatibility should be considered in organisations during the recruitment process, selection process, and throughout job design.
*&lt;u&gt;Kristof-Brown et al. (2005):&lt;/u&gt;
**Empirical evidence through a meta-analysis, which supports the importance of different 'fit' forms.
**Encourages organisations to consider compatibility through organisational values but also through jobs, teams, and supervisors. 
*&lt;u&gt;Lee et al. (2025):&lt;/u&gt;
**Emphasises how leader-follower value congruence is important in sustaining workplace dynamics and relationships.
**Managers can support motivation through developing stronger connections between employee values and leader values.
*&lt;u&gt;Wang et al. (2018):&lt;/u&gt;
**Leadership can connect organisational values with employee behaviour.
**Discusses how leaders can model and reinforce shared values to encourage motivated behaviour.

=== '''Education and Everyday Life Value Congruence''' ===
* &lt;u&gt;Ryan &amp; Deci (2018):&lt;/u&gt;
** Applies value congruence beyond workplaces, especially in regard to autonomous motivation and meaningful, personal goals.
** Education example: activities that align with student values/goals may motivate students to learn more effectively and productively.
*&lt;u&gt;Schwartz (2012):&lt;/u&gt;
**Individual differences can explain why different students or individuals may respond in different ways to the same situation - they prioritise different values.
**Applied in both educational and everyday decision-making - considers how goals/tasks may reflect personal values.
*&lt;u&gt;van Vianen (2018):&lt;/u&gt;
**The person-environment fit framework can be applied outside organisational contexts to best understand how individuals react and interact with different environments.
**Can argue that motivation is supported when environments (e.g., schools, social groups, everyday settings) align with individual characteristics/values.


==Learning features==

=== Activity 1 - Motivation ===
Watch the linked youtube video before beginning Activity 1: https://www.youtube.com/watch?v=rrkrvAUbU9Y
;{{RoundBoxTop|theme=3}}'''The Candle Scenario''' 
&lt;br&gt;
In his TED talk, Dan Pink talks discusses motivation through &quot;The Candle Scenario&quot;. Pink introduces the candle problem, originally developed by psychologist Karl Duncker in 1945. 
&lt;br&gt;
Participants are given:
 - a candle
 - a box of thumbtacks
 - some matches
 - a wall
 - a table
Answer the following multiple choice questions based on this scenario.{{RoundBoxBottom}}&lt;quiz display=&quot;simple&quot;&gt;
{What was the main task in the candle problem?
|type=&quot;()&quot;}
+ Attach the candle to the wall without wax dripping onto the table.
- Light the candle without using matches.
- Use the thumbtacks to build a stand for the candle.
- Attach the candle directly to the table using the thumbtacks.

{Why did participants initially struggle to solve the candle problem?
|type=&quot;()&quot;}
+ They viewed the box only as a container for the thumbtacks rather than recognising that it could be used as part of the solution.
- They did not have enough thumbtacks to attach the candle to the wall.
- They were unable to light the candle because the matches were too short.
- They believed the candle could not be removed from its packaging.

{What did the candle problem demonstrate about the effects of external rewards on motivation?
|type=&quot;()&quot;}
+ External rewards can sometimes interfere with creative problem-solving on complex tasks.
- External rewards always increase motivation and improve creative performance.
- External rewards have no relationship with performance on creative tasks.
- External rewards are the only effective way to motivate people to solve difficult problems.
&amp;lt;/quiz&gt;
&lt;/quiz&gt;

=== Activity 2 - Value Congruence ===
The following scenario and subsequent quiz are designed to test the reader's understanding of value congruence and motivation in everyday life settings.

&lt;nowiki&gt;*&lt;/nowiki&gt;''Authors Note: Will insert scenario and quiz for activity 2 at a later stage once chapter is more developed''.*
;{{RoundBoxTop|theme=3}}'''Scenario 2:''' 
&lt;br&gt;
{{RoundBoxBottom}}&lt;quiz display=&quot;simple&quot;&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;
==Conclusion==
* The conclusion will bring the attention back to the scenario described in the overview, however, it will answer the fundamental question of why yourself (in the scenario) and Jane experience such differences in motivation.
* Will review each of the focus questions and present a finalised answer based on the chapter content.
* Will summarise how value congruence is related to motivation through theory and empirical evidence.
*
* 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==

* [[wikipedia:Motivation|Motivation]] (Wikipedia)
* [[Motivation and emotion/Book/2025/Theory of basic human values|Theory of basic human values]] (Book chapter, 2025)
* [[wikipedia:Value_(ethics)|Values (ethics)]] (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=

Bao, Y., Vedina, R., Moodie, S., &amp; Dolan, S. (2013). The relationship between value incongruence and individual and organizational well-being outcomes: An exploratory study among Catalan nurses. Journal of Advanced Nursing, 69(3), 631–641. https://doi.org/10.1111/j.1365-2648.2012.06045.x

Hoffman, B. J., &amp; Woehr, D. J. (2006). A quantitative review of the relationship between person–organization fit and behavioral outcomes. Journal of Vocational Behavior, 68(3), 389–399. https://doi.org/10.1016/j.jvb.2005.08.003

Hossli, N., Natter, M., &amp; Algesheimer, R. (2025). On the importance of congruence between personal and work values – How value incongruence affects job satisfaction: A multiple mediation model. International Journal of Wellbeing, 14(3), 1–18. https://doi.org/10.5502/ijw.v14i3.2905

Kristof, A. L. (1996). Person-organization fit: An integrative review of its conceptualizations, measurement, and implications. Personnel Psychology, 49(1), 1–49. https://doi.org/10.1111/j.1744-6570.1996.tb01790.x

Kristof-Brown, A. L., Zimmerman, R. D., &amp; Johnson, E. C. (2005). Consequences of individuals fit at work: A meta-analysis of person-job, person-organization, person-group, and person-supervisor fit. Personnel Psychology, 58(2), 281–342. https://doi.org/10.1111/j.1744-6570.2005.00672.x

Lee, A., Newman, A., Maio, G. R., Piplani, R. S., &amp; Harvey, W. S. (2025). Leader-follower value congruence: A systematic review of the literature and a future research agenda. The Leadership Quarterly, 36(6), Article 101922. https://doi.org/10.1016/j.leaqua.2025.101922

Levontin, L., &amp; Bardi, A. (2019). Using personal values to understand the motivational basis of amity goal orientation. Frontiers in Psychology, 9, 2736. https://doi.org/10.3389/fpsyg.2018.02736

Li, M., Wang, Z., You, X., &amp; Gao, J. (2015). Value congruence and teachers’ work engagement: The mediating role of autonomous and controlled motivation. Personality and Individual Differences, 80, 113–118. https://doi.org/10.1016/j.paid.2015.02.021

Rawolle, M., Wallis, M. S. v, Badham, R., &amp; Kehr, H. M. (2016). No fit, no fun: The effect of motive incongruence on job burnout and the mediating role of intrinsic motivation. Personality and Individual Differences, 89, 65–68. https://doi.org/10.1016/j.paid.2015.09.030

Ryan, R. M., &amp; Deci, E. L. (2018). Self-determination theory: Basic psychological needs in motivation, development, and wellness. The Guildford Press.

Schwartz, S. H. (2012). An overview of the Schwartz theory of basic values. Online Readings in Psychology and Culture, 2(1). https://doi.org/10.9707/2307-0919.1116

van Vianen, A. E. (2018). Person–environment fit: A review of its basic tenets. Annual Review of Organizational Psychology and Organizational Behavior, 5(5), 75-101. https://doi.org/10.1146/annurev-orgpsych-032117-104702

Wang, X., Zhou, K., &amp; Liu, W. (2018). Value Congruence: A Study of Green Transformational Leadership and Employee Green Behavior. Frontiers in Psychology, 9, 1946. https://doi.org/10.3389/fpsyg.2018.01946

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* 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://www.melrobbins.com/topic/motivation/ The motivation hub] - (Mel Robbins Podcast)
* [https://ebookcentral.proquest.com/lib/canberra/detail.action?docID=6488159 Understanding motivation and emotion] - (Reeve, 2024) {{ic|If using this as a source, move into References}}
* [https://www.ted.com/talks/anna_marie_dipasquale_align_your_life_with_your_core_values_and_natural_talents Value congruence TED talk] - (DiPasquale, 2019)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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__TOC__

==Overview==
{{RoundBoxTop|theme=4}}
[[File:WORKOUT.jpg|thumb|Man earning points for exercising (figure 1)]] 

'''Case study: A little more motivation'''

Alex has recently started using a fitness app to become more physically active. The app awards points for completing workouts, gives badges for reaching milestones, displays weekly challenges, and allows users to compare their progress with friends (see figure 1). At first, Alex exercises mainly to improve their health, but soon becomes motivated by earning enough points to reach the next level. When Alex moves up the leaderboard, they feel a sense of achievement and become more motivated to exercise again. However, after falling behind other users, Alex begins to lose interest and considers stopping.                                   

Alex's experience raises an important question: why can the same gamification features that motivate exercise for some people have different effects on motivation and behaviour for others?                                                    
{{RoundBoxBottom}}

Regular physical activity provides important physical and psychological benefits, yet maintaining exercise behaviour can be difficult. Digital fitness technologies increasingly use gamification, such as points, rewards, challenges, feedback and social comparison, to encourage people to become more active. Gamification may make exercise more engaging and provide users with goals and feedback, but its effects on motivation and behaviour are not necessarily straightforward.

Psychological science can help explain why gamification may influence exercise motivation and whether increased motivation translates into sustained exercise behaviour. Understanding theories of motivation can also help identify when gamification is likely to support engagement and when it may instead reduce motivation or discourage participation.

{{RoundBoxTop|theme=3}}
'''Focus questions'''

*What is exercise gamification and how is it used to influence motivation?
*What psychological mechanisms explain how gamification can affect exercise motivation?
*How does gamification influence actual exercise behaviour and adherence?
*When might gamification support or undermine exercise motivation?
*How can gamification be designed to encourage sustainable exercise behaviour?
{{RoundBoxBottom}}

== Understanding Exercise Gamification ==

* Gamification involves incorporating game-design elements into non-game contexts, such as fitness apps and physical-activity programs, rather than turning exercise itself into a complete game (Krath et al., 2021).
* Common exercise gamification features include points, badges, rewards, challenges, progress tracking, feedback and leaderboards, which can provide users with goals and information about their progress (Shameli et al., 2017).
* Gamification is increasingly used in digital physical-activity interventions, although research indicates that the effectiveness of different game elements can vary across interventions and populations (Wang et al., 2025).
* Gamification can be understood as a potential behaviour-change strategy, making it important to examine the psychological mechanisms through which game elements may influence motivation and exercise behaviour (Mazeas et al., 2022).

== Psychological Mechanisms of Gamification ==

* Self-Determination Theory (SDT) proposes that motivation is influenced by the satisfaction of three basic psychological needs: autonomy, competence and relatedness. Different forms of motivation range from autonomous/intrinsic motivation to controlled/extrinsic motivation (Ryan &amp; Deci, 2020).
* Gamification features such as progress feedback, achievable challenges and goals may support feelings of competence by helping users recognise improvement and successful performance.
* Social features such as leaderboards, competition and shared challenges may influence relatedness and social comparison, although their effects may depend on the individual and how the feature is designed (Karabiber &amp; Gürol, 2026).
* External rewards and points may encourage behaviour, but gamification should not be assumed to automatically produce autonomous motivation; the psychological quality of the motivation matters.
* Research applying SDT to physical activity generally finds positive relationships between more self-determined forms of motivation and physical activity, while controlled motivation and amotivation show weaker or negative relationships (Teixeira et al., 2012).

== Gamification and Exercise Behaviour ==

* Increased motivation does not necessarily mean that a person will maintain exercise behaviour, so research needs to distinguish between motivation, engagement and actual physical activity.
* A systematic review and meta-analysis of 16 randomised controlled trials involving 2,407 participants found that gamified interventions had a small-to-medium positive effect on physical activity, although the effect was weaker at longer-term follow-up (Mazeas et al., 2021). 
* Research on gamified smartphone applications similarly indicates small-to-moderate improvements in physical activity but the large variation between studies suggests that the effectiveness of gamification depends partly on how interventions are designed (Yang et al., 2021). 
* Long-term exercise adherence should therefore be considered separately from short-term increases in engagement, especially because gamification effects may become smaller over time. 

== Designing Effective and Sustainable Exercise Gamification ==

* Effective gamification should be based on psychological principles and behaviour-change evidence, rather than simply adding game elements such as points or badges.
* Features that provide meaningful goals, feedback and opportunities for progress may help support motivation, particularly when they encourage feelings of competence and autonomy (Krath et al., 2021). 
* Competitive features such as leaderboards may motivate some users but could discourage others, highlighting the importance of individual differences and the way gamification features are implemented.
* Gamification should aim to support sustained exercise behaviour, rather than relying only on novelty, rewards or short-term engagement.
* Future exercise-gamification interventions should investigate which combinations of game elements are most effective for the majority of people as current research shows considerable variation between interventions (Xu et al., 2022). 
&lt;quiz display=simple&gt; 
'''Which feature is most likely to support autonomous motivation?'''}

- Users lose points whenever they miss a workout.
+ Users choose their own exercise goals and receive feedback about their progress.
- Users are publicly ranked against other users.
&lt;/quiz&gt;

==Learning features==
A short case study will compare two people who respond differently to the same gamified fitness app and ask readers to explain the difference using motivational theory.

A feature box will also be added to summarise key evidence-based principles for using gamification to support sustainable exercise motivation

==Conclusion==
* Gamification can influence exercise motivation and behaviour by incorporating game elements such as goals, feedback, rewards, progress tracking and social interaction that can make exercise more engaging.
* Psychological theories, especially Self-Determination Theory, can help explain why gamification may strengthen or weaken motivation depending on how game elements support or frustrate psychological needs such as autonomy, competence and relatedness.
* Research suggests that gamification can increase physical activity and engagement, but effects vary between individuals and may decrease over time, highlighting the importance of considering long-term exercise adherence.
* Effective exercise gamification should support meaningful and sustainable motivation rather than relying solely on external rewards. Practical strategies may include providing meaningful choices, achievable challenges, progress feedback and opportunities for social connection while avoiding excessive pressure or competition.

==See also==

*[[Motivation and emotion/Book/2017/Gamification and motivation|Gamification and motivation]] (Book chapter, 2017)
*[[Motivation and emotion/Book/2021/Physical activity motivation|Physical activity motivation]] (Book chapter, 2021)
*[[Motivation and emotion/Book/2023/Physical activity tracking and exercise motivation|Physical activity tracking and exercise motivation]] (Book chapter, 2023)
*[https://en.wikipedia.org/wiki/Self-determination_theory Self-Determination Theory] (Wikipedia) {{ic|Use internal link style as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Krath, J., Schürmann, L., &amp; Von Korflesch, H. F. (2021). Revealing the theoretical basis of gamification: A systematic review and analysis of theory in research on gamification, serious games and game-based learning. Computers in Human Behavior, 125, 106963. https://doi.org/10.1016/j.chb.2021.106963

Shameli, A., Althoff, T., Saberi, A., &amp; Leskovec, J. (2017). How Gamification Affects Physical Activity. Proc Int World Wide Web Conf, 2017, 455–463. https://doi.org/10.1145/3041021.3054172

Wang, M., Xu, J., Zhou, X., Li, X., &amp; Zheng, Y. (2025). Effectiveness of Gamification Interventions to Improve Physical activity and Sedentary Behavior in Children and Adolescents: Systematic Review and Meta-Analysis. JMIR Serious Games, 13, e68151. https://doi.org/10.2196/68151

Mazeas, A., Duclos, M., Pereira, B., &amp; Chalabaev, A. (2021). Evaluating the effectiveness of gamification on physical activity: systematic review and meta-analysis of randomized controlled trials. Journal of Medical Internet Research, 24(1), e26779. https://doi.org/10.2196/26779

Ryan, R. M., &amp; 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

Karabiber, H., &amp; Gürol, Y. D. (2026). Feedback, competition, and cooperation: Behavioral effects of gamification elements in individual and team settings. Acta Psychologica, 266, 106908. https://doi.org/10.1016/j.actpsy.2026.106908

Teixeira, P. J., Carraça, E. V., Markland, D., Silva, M. N., &amp; Ryan, R. M. (2012). Exercise, physical activity, and self-determination theory: A systematic review. International Journal of Behavioral Nutrition and Physical Activity, 9(1), 78. https://doi.org/10.1186/1479-5868-9-78

Yang, Y., Hu, H., &amp; Koenigstorfer, J. (2021). Effects of Gamified Smartphone Applications on Physical Activity: A Systematic Review and Meta-Analysis. American Journal of Preventive Medicine, 62(4), 602–613. https://doi.org/10.1016/j.amepre.2021.10.005

Xu, L., Shi, H., Shen, M., Ni, Y., Zhang, X., Pang, Y., Yu, T., Lian, X., Yu, T., Yang, X., &amp; Li, F. (2022). The Effects of MHealth-Based Gamification Interventions on Participation in Physical Activity: Systematic review. JMIR Mhealth and Uhealth, 10(2), e27794. https://doi.org/10.2196/27794

}}

==External links==
{{ic|Use bullet points}}

[https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians?language=en 24-hour movement guidelines for all Australians] (Australian Government Department of Health, Disability and Ageing)

[https://www.who.int/news-room/fact-sheets/detail/physical-activity Physical activity] (World Health Organization)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Exercise]]
[[Category:Motivation and emotion/Book/Gamification]]</text>
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      <text bytes="11802" sha1="dffuk60umuij4m0hf6os8shwhp7ml5h" xml:space="preserve">{{title|Exercise gamification motivation:&lt;br&gt;How can gamification affect exercise motivation and behaviour?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=4}}
[[File:WORKOUT.jpg|thumb|Man earning points for exercising (figure 1)]] 

'''Case study: A little more motivation'''

Alex has recently started using a fitness app to become more physically active. The app awards points for completing workouts, gives badges for reaching milestones, displays weekly challenges, and allows users to compare their progress with friends (see figure 1). At first, Alex exercises mainly to improve their health, but soon becomes motivated by earning enough points to reach the next level. When Alex moves up the leaderboard, they feel a sense of achievement and become more motivated to exercise again. However, after falling behind other users, Alex begins to lose interest and considers stopping.                                   

Alex's experience raises an important question: why can the same gamification features that motivate exercise for some people have different effects on motivation and behaviour for others?                                                    
{{RoundBoxBottom}}

Regular physical activity provides important physical and psychological benefits, yet maintaining exercise behaviour can be difficult. Digital fitness technologies increasingly use gamification, such as points, rewards, challenges, feedback and social comparison, to encourage people to become more active. Gamification may make exercise more engaging and provide users with goals and feedback, but its effects on motivation and behaviour are not necessarily straightforward.

Psychological science can help explain why gamification may influence exercise motivation and whether increased motivation translates into sustained exercise behaviour. Understanding theories of motivation can also help identify when gamification is likely to support engagement and when it may instead reduce motivation or discourage participation.

{{RoundBoxTop|theme=3}}
'''Focus questions'''

*What is exercise gamification and how is it used to influence motivation?
*What psychological mechanisms explain how gamification can affect exercise motivation?
*How does gamification influence actual exercise behaviour and adherence?
*When might gamification support or undermine exercise motivation?
*How can gamification be designed to encourage sustainable exercise behaviour?
{{RoundBoxBottom}}

== Understanding Exercise Gamification ==

* Gamification involves incorporating game-design elements into non-game contexts, such as fitness apps and physical-activity programs, rather than turning exercise itself into a complete game (Krath et al., 2021).
* Common exercise gamification features include points, badges, rewards, challenges, progress tracking, feedback and leaderboards, which can provide users with goals and information about their progress (Shameli et al., 2017).
* Gamification is increasingly used in digital physical-activity interventions, although research indicates that the effectiveness of different game elements can vary across interventions and populations (Wang et al., 2025).
* Gamification can be understood as a potential behaviour-change strategy, making it important to examine the psychological mechanisms through which game elements may influence motivation and exercise behaviour (Mazeas et al., 2022).

== Psychological Mechanisms of Gamification ==

* Self-Determination Theory (SDT) proposes that motivation is influenced by the satisfaction of three basic psychological needs: autonomy, competence and relatedness. Different forms of motivation range from autonomous/intrinsic motivation to controlled/extrinsic motivation (Ryan &amp; Deci, 2020).
* Gamification features such as progress feedback, achievable challenges and goals may support feelings of competence by helping users recognise improvement and successful performance.
* Social features such as leaderboards, competition and shared challenges may influence relatedness and social comparison, although their effects may depend on the individual and how the feature is designed (Karabiber &amp; Gürol, 2026).
* External rewards and points may encourage behaviour, but gamification should not be assumed to automatically produce autonomous motivation; the psychological quality of the motivation matters.
* Research applying SDT to physical activity generally finds positive relationships between more self-determined forms of motivation and physical activity, while controlled motivation and amotivation show weaker or negative relationships (Teixeira et al., 2012).

== Gamification and Exercise Behaviour ==

* Increased motivation does not necessarily mean that a person will maintain exercise behaviour, so research needs to distinguish between motivation, engagement and actual physical activity.
* A systematic review and meta-analysis of 16 randomised controlled trials involving 2,407 participants found that gamified interventions had a small-to-medium positive effect on physical activity, although the effect was weaker at longer-term follow-up (Mazeas et al., 2021). 
* Research on gamified smartphone applications similarly indicates small-to-moderate improvements in physical activity but the large variation between studies suggests that the effectiveness of gamification depends partly on how interventions are designed (Yang et al., 2021). 
* Long-term exercise adherence should therefore be considered separately from short-term increases in engagement, especially because gamification effects may become smaller over time. 

== Designing Effective and Sustainable Exercise Gamification ==

* Effective gamification should be based on psychological principles and behaviour-change evidence, rather than simply adding game elements such as points or badges.
* Features that provide meaningful goals, feedback and opportunities for progress may help support motivation, particularly when they encourage feelings of competence and autonomy (Krath et al., 2021). 
* Competitive features such as leaderboards may motivate some users but could discourage others, highlighting the importance of individual differences and the way gamification features are implemented.
* Gamification should aim to support sustained exercise behaviour, rather than relying only on novelty, rewards or short-term engagement.
* Future exercise-gamification interventions should investigate which combinations of game elements are most effective for the majority of people as current research shows considerable variation between interventions (Xu et al., 2022). 

&lt;quiz display=simple&gt; 
'''Which feature is most likely to support autonomous exercise motivation?'''}

- Users lose points whenever they miss a workout.
+ Users choose their own exercise goals and receive feedback about their progress.
- Users are publicly ranked against other users.
&lt;/quiz&gt;

==Learning features==
A short case study will compare two people who respond differently to the same gamified fitness app and ask readers to explain the difference using motivational theory.

A feature box will also be added to summarise key evidence-based principles for using gamification to support sustainable exercise motivation

==Conclusion==
* Gamification can influence exercise motivation and behaviour by incorporating game elements such as goals, feedback, rewards, progress tracking and social interaction that can make exercise more engaging.
* Psychological theories, especially Self-Determination Theory, can help explain why gamification may strengthen or weaken motivation depending on how game elements support or frustrate psychological needs such as autonomy, competence and relatedness.
* Research suggests that gamification can increase physical activity and engagement, but effects vary between individuals and may decrease over time, highlighting the importance of considering long-term exercise adherence.
* Effective exercise gamification should support meaningful and sustainable motivation rather than relying solely on external rewards. Practical strategies may include providing meaningful choices, achievable challenges, progress feedback and opportunities for social connection while avoiding excessive pressure or competition.

==See also==

*[[Motivation and emotion/Book/2017/Gamification and motivation|Gamification and motivation]] (Book chapter, 2017)
*[[Motivation and emotion/Book/2021/Physical activity motivation|Physical activity motivation]] (Book chapter, 2021)
*[[Motivation and emotion/Book/2023/Physical activity tracking and exercise motivation|Physical activity tracking and exercise motivation]] (Book chapter, 2023)
*[https://en.wikipedia.org/wiki/Self-determination_theory Self-Determination Theory] (Wikipedia) {{ic|Use internal link style as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Krath, J., Schürmann, L., &amp; Von Korflesch, H. F. (2021). Revealing the theoretical basis of gamification: A systematic review and analysis of theory in research on gamification, serious games and game-based learning. Computers in Human Behavior, 125, 106963. https://doi.org/10.1016/j.chb.2021.106963

Shameli, A., Althoff, T., Saberi, A., &amp; Leskovec, J. (2017). How Gamification Affects Physical Activity. Proc Int World Wide Web Conf, 2017, 455–463. https://doi.org/10.1145/3041021.3054172

Wang, M., Xu, J., Zhou, X., Li, X., &amp; Zheng, Y. (2025). Effectiveness of Gamification Interventions to Improve Physical activity and Sedentary Behavior in Children and Adolescents: Systematic Review and Meta-Analysis. JMIR Serious Games, 13, e68151. https://doi.org/10.2196/68151

Mazeas, A., Duclos, M., Pereira, B., &amp; Chalabaev, A. (2021). Evaluating the effectiveness of gamification on physical activity: systematic review and meta-analysis of randomized controlled trials. Journal of Medical Internet Research, 24(1), e26779. https://doi.org/10.2196/26779

Ryan, R. M., &amp; 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

Karabiber, H., &amp; Gürol, Y. D. (2026). Feedback, competition, and cooperation: Behavioral effects of gamification elements in individual and team settings. Acta Psychologica, 266, 106908. https://doi.org/10.1016/j.actpsy.2026.106908

Teixeira, P. J., Carraça, E. V., Markland, D., Silva, M. N., &amp; Ryan, R. M. (2012). Exercise, physical activity, and self-determination theory: A systematic review. International Journal of Behavioral Nutrition and Physical Activity, 9(1), 78. https://doi.org/10.1186/1479-5868-9-78

Yang, Y., Hu, H., &amp; Koenigstorfer, J. (2021). Effects of Gamified Smartphone Applications on Physical Activity: A Systematic Review and Meta-Analysis. American Journal of Preventive Medicine, 62(4), 602–613. https://doi.org/10.1016/j.amepre.2021.10.005

Xu, L., Shi, H., Shen, M., Ni, Y., Zhang, X., Pang, Y., Yu, T., Lian, X., Yu, T., Yang, X., &amp; Li, F. (2022). The Effects of MHealth-Based Gamification Interventions on Participation in Physical Activity: Systematic review. JMIR Mhealth and Uhealth, 10(2), e27794. https://doi.org/10.2196/27794

}}

==External links==
{{ic|Use bullet points}}

[https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians?language=en 24-hour movement guidelines for all Australians] (Australian Government Department of Health, Disability and Ageing)

[https://www.who.int/news-room/fact-sheets/detail/physical-activity Physical activity] (World Health Organization)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Exercise]]
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      <text bytes="4624" sha1="komwk7sz5t8ljdhkc4y6mo5zeq0ozjs" xml:space="preserve">== About Me ==
I am a military veteran, working to complete a Bachelor of Science in Psychology as part of a lifelong approach to learning. After a military career working with people around the world on a spectrum of operations, I wanted to better understand the human dimension and the science behind what makes us all tick. Along the way, I want to give back to the community where I can. I have an interest in veterans' PTSD, advocacy and international affairs.

== Book Chapter ==
[[File:PTSD.png|left|thumb|250x250px|'''Figure 1.''' PTSD can have a deep and lasting impact on our emotions.]]
I am writing a book chapter titled [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|Immersive Therapy for PTSD: How does it work and what are the effects?]] 

This chapter is an individual student contribution as part of a larger [https://www.canberra.edu.au/ University of Canberra] class effort in 2026 to write a book [[Motivation and emotion/Book]]. 

The 2026 book extends a Wikiversity book series started in 2010, with over 1,800 online book chapters on how psychological science can improve human lives.

The book editor and University psychology unit convener is [[User:Jtneill|Dr. James Neill]]
{{clear}}

== Social Contributions ==

# '''18 Aug 26:''' Grammar and content editing improvements on 2025 chapter, Defence mechanisms and emotion regulation. Edits at https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FDefence_mechanisms_and_emotion_regulation&amp;diff=2822629&amp;oldid=2763661
# '''18 Aug 26:''' Grammar and content editing improvements on 2025 chapter, Neurodivergence and trauma. Edits at https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FNeurodivergence_and_trauma&amp;diff=2822636&amp;oldid=2812805
# '''19 Aug 26:''' Grammar and content editing improvements on 2025 chapter, Incentive theory of motivation. Edits at&lt;nowiki/&gt;https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FIncentive_theory_of_motivation&amp;diff=2822844&amp;oldid=2814898
# '''19 Aug 26:''' UC Canvas Discussion: Introduced the book chapter intent to the class and welcomed anyone with sim&lt;nowiki/&gt;ilar topics or interests to contribute. I offered the same. https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456450
# '''20 Aug 26:''' Comment: Supported another Wiki user by providing some peer-reviewed ideas on starter references f&lt;nowiki/&gt;or a Chapter page. [[Talk:Motivation and emotion/Book/2026/Perfectionism and procrastination#Reference material]]
# '''20 Aug 26:''' Comment: Supported another Wiki user by providing some peer-reviewed ideas on starter references f&lt;nowiki/&gt;or a Chapter page.[[Talk:Motivation and emotion/Book/2026/Irritability#Reference material]]
# '''22 Aug 26:''' Grammar and content editing improvements on 2025 chapter, Mental health in astronauts. Edits at [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FMental_health_in_astronauts&amp;diff=2824159&amp;oldid=2804667 ht]&lt;nowiki/&gt;[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FMental_health_in_astronauts&amp;diff=2824159&amp;oldid=2804667 tps://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FMental_health_in_astronauts&amp;diff=2824159&amp;oldid=2804667]
# '''25 Aug 26:''' Comment: Supported another Wiki user by providing some peer-reviewed ideas on starter references f&lt;nowiki/&gt;or a Chapter page.[[Talk:Motivation and emotion/Book/2026/Feedback literacy#Reference]]
# '''25 Aug 26''': UC Canvas Discussion: Responded to student topic query.[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457190 https://uclearn.canberra.edu.au/courses/201]&lt;nowiki/&gt;[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457190 43/discussion_topics/457190]
# '''26 Aug 26:''' Comment: Supported another Wiki user by providing some external link ideas for a Chapter page.[[Talk:Motivation and emotion/Book/2026/Dark empathy#Reference|Talk]]&lt;nowiki/&gt;[[Talk:Motivation and emotion/Book/2026/Dark empathy#Reference|:Motivation and emotion/Book/2026/Dark empathy#Reference]]
# '''26 Aug 26: UC Canvas Discussion:''' Responded to student topic query with data from a Random Game of Dice scenar&lt;nowiki/&gt;io.https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457412
# '''27 Aug 26:''' Wiki thanks and editing conversation: [[User talk:Jtneill#Punctuation change in book title]]
# '''28 Aug 26:''' '''UC Canvas Discussion:''' Responded to student topic query&lt;nowiki/&gt;https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457809</text>
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      <text bytes="10110" sha1="9nxbpcr50u0ssjxiuwwsl562yz6rp1t" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Use a captioned image to illustrate the scenario]]

; Imagine this ... or Scenario ... or Case study or ... ?)

Start with an engaging [[#Scenarios|scenario, example, or case study]] which illustrates the problem and engages reader interest.

Present the scenario in a [[#Feature box|feature box]]. To change the box colour:
# Edit source
# Change &quot;theme=3&quot; to another number

Include an image and cite it (e.g., see Figure 1).

{{RoundBoxBottom}}

The Overview section should provide:
# '''Scenario''': A short, engaging case study or real-world example in a feature box, with an accompanying image (see above)
# '''Explanation of the problem, issue, or topc''': Briefly explain the problem, why it is important, and outline how psychological science can help
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box

Recommended length: 180 to 330 words.

This template provides key headings, examples, and tips for each section. Gradually remove this generic information as the chapter develops. It is OK to retain some template material for the topic development, but it should all be removed from the book chapter.

Key resources:
* [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] explains how to edit
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]

{{RoundBoxTop|theme=3}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* What is the first focus question?
* What is the second focus question? 
* What is the third focus question?

Ask [[w:Open-ended question|open-ended]] questions. For example:

{{cross}} Is there a relationship between weather and criminal behaviour? (closed-ended)&lt;br&gt;{{tick}} What is the relationship between weather and criminal behaviour? (open-ended)
{{RoundBoxBottom}}

==Headings==
Use this heading structure:
* [[#Overview|Overview]]
* 3 to 6 major headings tailored to the topic; can have sub-headings:
** avoid sections with only one sub-heading (use 0 or 2+ sub-headings)
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* [[#See also|See also]]
* [[#References|References]]
* [[#External links|External links]]

=Overview=

=Dark Empathy=
==Types of Empathy==
Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others&lt;p&gt;There are two types of empathy: &lt;p&gt; * Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain &lt;p&gt; * Affective empathy - physical and emotional sharing of feelings &lt;/p&gt;

==The Dark Triad==
==Making of a Dark Empath==

=Consequences of Dark Empathy=
==Effects on Victims of Dark Empathy==
==Outcome for Dark Empaths==

=Address Dark Empathy=
==Regaining themselves: Recovering from a Dark Empath==
==Unravelling from Dark Empathy==

=Conclusion=

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; 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 &amp; Row.

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>9nxbpcr50u0ssjxiuwwsl562yz6rp1t</sha1>
    </revision>
    <revision>
      <id>2829811</id>
      <parentid>2829809</parentid>
      <timestamp>2026-08-31T02:01:10Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* Overview */ adding in dotpoints to shape overview</comment>
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      <model>wikitext</model>
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      <text bytes="9356" sha1="dgb1dn5frccqcruyka72ks5lquya44y" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
==Case Study==
*A friend discussing their concerns about their friend's long term relationship
*They are concerned about the relationship and the person having too much control
*The relationship has over taken their life and their partner is slowly diminishing their confidence and self worth

{{RoundBoxBottom}}

* Dark Empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas could assist with understanding the motivation behind dark empat
{{RoundBoxTop|theme=3}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* What is the first focus question?
* What is the second focus question? 
* What is the third focus question?

Ask [[w:Open-ended question|open-ended]] questions. For example:

{{cross}} Is there a relationship between weather and criminal behaviour? (closed-ended)&lt;br&gt;{{tick}} What is the relationship between weather and criminal behaviour? (open-ended)
{{RoundBoxBottom}}


==Headings==
Use this heading structure:
* [[#Overview|Overview]]
* 3 to 6 major headings tailored to the topic; can have sub-headings:
** avoid sections with only one sub-heading (use 0 or 2+ sub-headings)
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* [[#See also|See also]]
* [[#References|References]]

=Overview=

=Dark Empathy=
==Types of Empathy==
Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others&lt;p&gt;There are two types of empathy: &lt;p&gt; * Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain &lt;p&gt; * Affective empathy - physical and emotional sharing of feelings &lt;/p&gt;

==The Dark Triad==
==Making of a Dark Empath==

=Consequences of Dark Empathy=
==Effects on Victims of Dark Empathy==
==Outcome for Dark Empaths==

=Address Dark Empathy=
==Regaining themselves: Recovering from a Dark Empath==
==Unravelling from Dark Empathy==

=Conclusion=

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; 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 &amp; Row.

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>dgb1dn5frccqcruyka72ks5lquya44y</sha1>
    </revision>
    <revision>
      <id>2829812</id>
      <parentid>2829811</parentid>
      <timestamp>2026-08-31T02:15:30Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>updating the headings and adding dot points under headings</comment>
      <origin>2829812</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9191" sha1="qrp8nx5b0tiuznohanswd430dm0e2tc" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
==Case study==
*A friend discussing their concerns about their friend's long term relationship
*They are concerned about the relationship and the person having too much control
*The relationship has over taken their life and their partner is slowly diminishing their confidence and self worth

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas could assist with understanding the motivation behind dark empathy

== Dark empathy ==

* Development of dark empathy '''i'''s when an individual has high levels of dark traits and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)
*

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others&lt;p&gt;There are two types of empathy: 

* Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain
* Affective empathy - physical and emotional sharing of feelings

=== The dark triad ===

* Narcisissm
* Machiavellianism
* Psychopathy

=== Making of a dark empath ===

* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy.
*

== Consequences of dark empathy ==

*

=== Effects on victims of dark empathy ===

*

=== Outcome for dark empaths ===

*

== Address dark empathy ==

*

=== Regaining themselves: recovering from a dark empath ===

*

=== Unravelling from Dark Empathy ===

*

== Conclusion ==

* Emotional intelligence does not come with morality

* The dark triad is an indicator of dark empaths if they have high dark traits and affective cognitive empathy
* Cognitive empathy and Trauma patterns can develop the drive for using dark empathy

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; 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 &amp; Row.

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>qrp8nx5b0tiuznohanswd430dm0e2tc</sha1>
    </revision>
    <revision>
      <id>2829815</id>
      <parentid>2829812</parentid>
      <timestamp>2026-08-31T02:32:26Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* The dark triad */ updating the dotpoint</comment>
      <origin>2829815</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9125" sha1="73fxt0y6rt59th64lk0xn0v70qkkppv" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
==Case study==
*A friend discussing their concerns about their friend's long term relationship
*They are concerned about the relationship and the person having too much control
*The relationship has over taken their life and their partner is slowly diminishing their confidence and self worth

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas could assist with understanding the motivation behind dark empathy

== Dark empathy ==

* Development of dark empathy '''i'''s when an individual has high levels of dark traits and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)
*

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others&lt;p&gt;There are two types of empathy: 

* Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain
* Affective empathy - physical and emotional sharing of feelings

=== The dark triad ===

* Narcissism
* Machiavellianism
* Psychopathy

=== Making of a dark empath ===

* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy.
*

== Consequences of dark empathy ==

*

=== Effects on victims of dark empathy ===

*

=== Outcome for dark empaths ===

*

== Address dark empathy ==

*

=== Regaining themselves: recovering from a dark empath ===

*

=== Unravelling from Dark Empathy ===

*

== Conclusion ==

* Emotional intelligence does not come with morality

* The dark triad is an indicator of dark empaths if they have high dark traits and affective cognitive empathy
* Cognitive empathy and Trauma patterns can develop the drive for using dark empathy

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>73fxt0y6rt59th64lk0xn0v70qkkppv</sha1>
    </revision>
    <revision>
      <id>2829868</id>
      <parentid>2829815</parentid>
      <timestamp>2026-08-31T10:30:01Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* Dark empathy */ updating with more information under the headings</comment>
      <origin>2829868</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10497" sha1="feeaaih2uscb3dpvs6793suuk6grtsu" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}
==Case study==
*A friend discussing their concerns about their friend's long term relationship
*They are concerned about the relationship and the person having too much control
*The relationship has over taken their life and their partner is slowly diminishing their confidence and self worth

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas could assist with understanding the motivation behind dark empathy

== Dark empathy ==

* [[Empathy]] is a shared experience of feelings and thoughts while dark empathy is understanding those feelings and harnessing against others
* Development of dark empathy '''i'''s when an individual has high levels of dark traits and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others&lt;p&gt;There are two types of empathy: 

* Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain
* Affective empathy - physical and emotional sharing of feelings

=== The dark triad ===

* There are three sides to the dark triad: 
** Narcissism
** Machiavellianism
** Psychopathy

=== Making of a dark empath ===

* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy.
*

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and maniuplation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised
*

=== Regaining themselves: recovering from a dark empath ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from Dark Empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for Dark Empaths are:
**Trauma-informed therapy
**CBT
**Schema therapy
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality

* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>feeaaih2uscb3dpvs6793suuk6grtsu</sha1>
    </revision>
    <revision>
      <id>2829870</id>
      <parentid>2829868</parentid>
      <timestamp>2026-08-31T10:34:54Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* Unravelling from Dark Empathy */ changing captialisation and grammar</comment>
      <origin>2829870</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10379" sha1="dfbet0ci62kkcfsbtks81uhpzonfzvk" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}
==Case study==
*A friend discussing their concerns about their friend's long term relationship
*They are concerned about the relationship and the person having too much control
*The relationship has over taken their life and their partner is slowly diminishing their confidence and self worth

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas could assist with understanding the motivation behind dark empathy

== Dark empathy ==

* [[Empathy]] is a shared experience of feelings and thoughts while dark empathy is understanding those feelings and harnessing against others
* Development of dark empathy '''i'''s when an individual has high levels of dark traits and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others&lt;p&gt;There are two types of empathy: 

* Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain
* Affective empathy - physical and emotional sharing of feelings

=== The dark triad ===

* There are three sides to the dark triad: 
** Narcissism
** Machiavellianism
** Psychopathy

=== Making of a dark empath ===

* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy.
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy
*

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and maniuplation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised
*

=== Regaining themselves: recovering from a dark empath ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for Dark Empaths are:
**Trauma-informed therapy
**CBT
**Schema therapy
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality

* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>dfbet0ci62kkcfsbtks81uhpzonfzvk</sha1>
    </revision>
    <revision>
      <id>2829872</id>
      <parentid>2829870</parentid>
      <timestamp>2026-08-31T10:41:07Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* Unravelling from dark empathy */ updating more dotpoints</comment>
      <origin>2829872</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10694" sha1="5isuqxzewsuhvhw3aa2p9g24lbppnem" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}
==Case study==
*A friend discussing their concerns about their friend's long term relationship
*They are concerned about the relationship and the person having too much control
*The relationship has over taken their life and their partner is slowly diminishing their confidence and self worth

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas could assist with understanding the motivation behind dark empathy

== Dark empathy ==

* [[Empathy]] is a shared experience of feelings and thoughts while dark empathy is understanding those feelings and harnessing against others
* Development of dark empathy '''i'''s when an individual has high levels of dark traits and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others&lt;p&gt;There are two types of empathy: 

* Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain
* Affective empathy - physical and emotional sharing of feelings

=== The dark triad ===

* There are three sides to the dark triad: 
** Narcissism
** Machiavellianism
** Psychopathy

=== Making of a dark empath ===

* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy.
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and maniuplation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empath ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**CBT
**Schema therapy
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality

* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
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    <revision>
      <id>2829874</id>
      <parentid>2829872</parentid>
      <timestamp>2026-08-31T10:45:24Z</timestamp>
      <contributor>
        <username>U3228742</username>
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      </contributor>
      <comment>/* Regaining themselves: recovering from a dark empath */ added a figure</comment>
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__TOC__
==Overview==
{{RoundBoxTop|theme=3}}
==Case study==
*A friend discussing their concerns about their friend's long term relationship
*They are concerned about the relationship and the person having too much control
*The relationship has over taken their life and their partner is slowly diminishing their confidence and self worth

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas could assist with understanding the motivation behind dark empathy

== Dark empathy ==

* [[Empathy]] is a shared experience of feelings and thoughts while dark empathy is understanding those feelings and harnessing against others
* Development of dark empathy '''i'''s when an individual has high levels of dark traits and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others&lt;p&gt;There are two types of empathy: 

* Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain
* Affective empathy - physical and emotional sharing of feelings
[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad develops self interest ]]

=== The dark triad ===

* There are three sides to the dark triad: 
** Narcissism
** Machiavellianism
** Psychopathy

=== Making of a dark empath ===

* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy.
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and manipulation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**CBT
**Schema therapy
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality

* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>lxloncy48iddldljotu5ss3kz4q6nqt</sha1>
    </revision>
    <revision>
      <id>2829877</id>
      <parentid>2829874</parentid>
      <timestamp>2026-08-31T10:51:34Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* Unravelling from dark empathy */</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11007" sha1="hyatpu8rtreaed7vkwffdyemq5x7nf7" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}
==Case study==
*A friend discussing their concerns about their friend's long term relationship
*They are concerned about the relationship and the person having too much control
*The relationship has over taken their life and their partner is slowly diminishing their confidence and self worth

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas could assist with understanding the motivation behind dark empathy

== Dark empathy ==

* [[Empathy]] is a shared experience of feelings and thoughts while dark empathy is understanding those feelings and harnessing against others
* Development of dark empathy '''i'''s when an individual has high levels of dark traits and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others&lt;p&gt;There are two types of empathy: 

* Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain
* Affective empathy - physical and emotional sharing of feelings
[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad develops self interest ]]

=== The dark triad ===

* There are three sides to the dark triad: 
** Narcissism
** Machiavellianism
** Psychopathy

=== Making of a dark empath ===

* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy.
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and manipulation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**[[w:Schema_therapy|Schema therapy]]
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality

* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
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      <contributor>
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        <id>3005570</id>
      </contributor>
      <comment>/* The dark triad */ updating figure description</comment>
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__TOC__
==Overview==
{{RoundBoxTop|theme=3}}
==Case study==
*A friend discussing their concerns about their friend's long term relationship
*They are concerned about the relationship and the person having too much control
*The relationship has over taken their life and their partner is slowly diminishing their confidence and self worth

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas could assist with understanding the motivation behind dark empathy

== Dark empathy ==

* [[Empathy]] is a shared experience of feelings and thoughts while dark empathy is understanding those feelings and harnessing against others
* Development of dark empathy '''i'''s when an individual has high levels of dark traits and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others&lt;p&gt;There are two types of empathy: 

* Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain
* Affective empathy - physical and emotional sharing of feelings
[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad and subsets of the three elements ]]

=== The dark triad ===

* There are three elements to the dark triad: 
** Narcissism
** Machiavellianism
** Psychopathy

=== Making of a dark empath ===

* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy.
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and manipulation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**[[w:Schema_therapy|Schema therapy]]
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality

* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
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      <text bytes="2176" sha1="hqank80leceb0s04emrndhq8xkaxan5" xml:space="preserve">== About Me ==
[[File:Noun psychology 1325504.svg|thumb|'''''Figure 1.''''' Symbol representative of studying psychology.]]
Hi Everyone!

I am a third year student (aged 20) at the [https://www.canberra.edu.au/ University of Canberra] completing an undergraduate degree in a Bachelor of Science in Psychology, with a Breadth Major in Human Resource Management. 

&lt;u&gt;Get to know me! I enjoy:&lt;/u&gt;

* Taking walks in nature
* Spending time with my dog 
* Partaking in retail therapy
* Interior design/decorating
* Learning about psychology

== My Book Chapter ==
As part of the unit [[Motivation and emotion|Motivation and Emotion (7124)]], I am working on a book chapter on [[Motivation and emotion/Book/2026/Value congruence and motivation|Value Congruence and Motivation]]. This chapter is centred around a core question; How does alignment between personal and situational values influence motivation?

Feel free to check out my chapter (as linked above)!

== Social Contributions ==

# [[:c:Student rushing to school.png|Uploaded an image (AI-Generated) of a student rushing to school to Wikimedia Commons through Upload Wizard for use on my book chapter overview.]]
# [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457190 Created a discussion page on UCLearn to share and gather opinions on my book chapter topic.]
#[https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2026/Self-disclosure_and_emotional_intimacy&amp;action=history Added a comment on another Wiki User's page suggesting a scenario they could use in their book chapter.]
#[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FExtended_process_model_of_emotion_regulation&amp;diff=2825013&amp;oldid=2821895 Corrected another Wiki User's title/sub-title to reflect the 2026 table of contents title/sub-title through a direct edit.]
#[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456861?entry_id=807869 Commented on a peer's discussion page about Wikimedia Commons, sharing my own use of it's image-upload feature.]
#[[c:File:Person-Organisation_Fit_Diagram.jpg|Uploaded a Person-Organisation Fit diagram to Wikimedia Commons.]]</text>
      <sha1>hqank80leceb0s04emrndhq8xkaxan5</sha1>
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      <comment>/* Social Contributions */</comment>
      <origin>2829801</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2175" sha1="d6gmbp8g4ce0x99hbj6hbb8wmg1klzg" xml:space="preserve">== About Me ==
[[File:Noun psychology 1325504.svg|thumb|'''''Figure 1.''''' Symbol representative of studying psychology.]]
Hi Everyone!

I am a third year student (aged 20) at the [https://www.canberra.edu.au/ University of Canberra] completing an undergraduate degree in a Bachelor of Science in Psychology, with a Breadth Major in Human Resource Management. 

&lt;u&gt;Get to know me! I enjoy:&lt;/u&gt;

* Taking walks in nature
* Spending time with my dog 
* Partaking in retail therapy
* Interior design/decorating
* Learning about psychology

== My Book Chapter ==
As part of the unit [[Motivation and emotion|Motivation and Emotion (7124)]], I am working on a book chapter on [[Motivation and emotion/Book/2026/Value congruence and motivation|Value Congruence and Motivation]]. This chapter is centred around a core question; How does alignment between personal and situational values influence motivation?

Feel free to check out my chapter (as linked above)!

== Social Contributions ==

# [[c:Student rushing to school.png|Uploaded an image (AI-Generated) of a student rushing to school to Wikimedia Commons through Upload Wizard for use on my book chapter overview.]]
# [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457190 Created a discussion page on UCLearn to share and gather opinions on my book chapter topic.]
#[https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2026/Self-disclosure_and_emotional_intimacy&amp;action=history Added a comment on another Wiki User's page suggesting a scenario they could use in their book chapter.]
#[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FExtended_process_model_of_emotion_regulation&amp;diff=2825013&amp;oldid=2821895 Corrected another Wiki User's title/sub-title to reflect the 2026 table of contents title/sub-title through a direct edit.]
#[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456861?entry_id=807869 Commented on a peer's discussion page about Wikimedia Commons, sharing my own use of it's image-upload feature.]
#[[c:File:Person-Organisation_Fit_Diagram.jpg|Uploaded a Person-Organisation Fit diagram to Wikimedia Commons.]]</text>
      <sha1>d6gmbp8g4ce0x99hbj6hbb8wmg1klzg</sha1>
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    <revision>
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      <contributor>
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      <format>text/x-wiki</format>
      <text bytes="2179" sha1="j0xpb0t4uofneyy4cpngvoed0usq7c4" xml:space="preserve">== About Me ==
[[File:Noun psychology 1325504.svg|thumb|'''''Figure 1.''''' Symbol representative of studying psychology.]]
Hi Everyone!

I am a third year student (aged 20) at the [https://www.canberra.edu.au/ University of Canberra] completing an undergraduate degree in a Bachelor of Science in Psychology, with a Breadth Major in Human Resource Management. 

&lt;u&gt;Get to know me! I enjoy:&lt;/u&gt;

* Taking walks in nature
* Spending time with my dog 
* Partaking in retail therapy
* Interior design/decorating
* Learning about psychology

== My Book Chapter ==
As part of the unit [[Motivation and emotion|Motivation and Emotion (7124)]], I am working on a book chapter on [[Motivation and emotion/Book/2026/Value congruence and motivation|Value Congruence and Motivation]]. This chapter is centred around a core question; How does alignment between personal and situational values influence motivation?

Feel free to check out my chapter (as linked above)!

== Social Contributions ==

# [[:File:Student rushing to school.png|Uploaded an image (AI-Generated) of a student rushing to school to Wikimedia Commons through Upload Wizard for use on my book chapter overview.]]
# [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457190 Created a discussion page on UCLearn to share and gather opinions on my book chapter topic.]
#[https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2026/Self-disclosure_and_emotional_intimacy&amp;action=history Added a comment on another Wiki User's page suggesting a scenario they could use in their book chapter.]
#[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FExtended_process_model_of_emotion_regulation&amp;diff=2825013&amp;oldid=2821895 Corrected another Wiki User's title/sub-title to reflect the 2026 table of contents title/sub-title through a direct edit.]
#[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456861?entry_id=807869 Commented on a peer's discussion page about Wikimedia Commons, sharing my own use of it's image-upload feature.]
#[[c:File:Person-Organisation_Fit_Diagram.jpg|Uploaded a Person-Organisation Fit diagram to Wikimedia Commons.]]</text>
      <sha1>j0xpb0t4uofneyy4cpngvoed0usq7c4</sha1>
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    <revision>
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      <comment>/* Social Contributions */ Fix link</comment>
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      <format>text/x-wiki</format>
      <text bytes="2178" sha1="mtk61stuu6tzxfbpzejsaoh9ibhq73o" xml:space="preserve">== About Me ==
[[File:Noun psychology 1325504.svg|thumb|'''''Figure 1.''''' Symbol representative of studying psychology.]]
Hi Everyone!

I am a third year student (aged 20) at the [https://www.canberra.edu.au/ University of Canberra] completing an undergraduate degree in a Bachelor of Science in Psychology, with a Breadth Major in Human Resource Management. 

&lt;u&gt;Get to know me! I enjoy:&lt;/u&gt;

* Taking walks in nature
* Spending time with my dog 
* Partaking in retail therapy
* Interior design/decorating
* Learning about psychology

== My Book Chapter ==
As part of the unit [[Motivation and emotion|Motivation and Emotion (7124)]], I am working on a book chapter on [[Motivation and emotion/Book/2026/Value congruence and motivation|Value Congruence and Motivation]]. This chapter is centred around a core question; How does alignment between personal and situational values influence motivation?

Feel free to check out my chapter (as linked above)!

== Social Contributions ==

# [[:File:Student rushing to school.png|Uploaded an image (AI-Generated) of a student rushing to school to Wikimedia Commons through Upload Wizard for use on my book chapter overview.]]
# [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457190 Created a discussion page on UCLearn to share and gather opinions on my book chapter topic.]
#[https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2026/Self-disclosure_and_emotional_intimacy&amp;oldid=2825009 Added a comment on another Wiki User's page suggesting a scenario they could use in their book chapter.]
#[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FExtended_process_model_of_emotion_regulation&amp;diff=2825013&amp;oldid=2821895 Corrected another Wiki User's title/sub-title to reflect the 2026 table of contents title/sub-title through a direct edit.]
#[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456861?entry_id=807869 Commented on a peer's discussion page about Wikimedia Commons, sharing my own use of it's image-upload feature.]
#[[c:File:Person-Organisation_Fit_Diagram.jpg|Uploaded a Person-Organisation Fit diagram to Wikimedia Commons.]]</text>
      <sha1>mtk61stuu6tzxfbpzejsaoh9ibhq73o</sha1>
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  <page>
    <title>Talk:Motivation and emotion/Book/2026/Exercise gamification motivation</title>
    <ns>1</ns>
    <id>331345</id>
    <revision>
      <id>2829791</id>
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      <timestamp>2026-08-30T23:01:35Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Topic development feedback</comment>
      <origin>2829791</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4527" sha1="0c7jy30dltwkpxb5bwhogqq9w3wz6ui" xml:space="preserve">== Overview ==

Looking good. I would suggest discussing some of the literature in this section. I say this because in the template it says to outline &quot;psychological science&quot;. [[User:Jack4234|Jack4234]] ([[User talk:Jack4234|discuss]] • [[Special:Contributions/Jack4234|contribs]]) 08:14, 20 August 2026 (UTC)

== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:U3260591|Hi [[User:U3260591|U3260591]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:01, 30 August 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings)
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- Focus questions --&gt;
# Reasonably good alignment between focus questions and heading structure, but consider closer alignment
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Solid development
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
# Select the best theories about this topic
# Select the best research about this topic
&lt;!-- Conclusion --&gt;
# Conclusion is underway
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Relevant figure(s) presented, captioned, and cited
&lt;!-- Caption --&gt;
# Figure caption(s) should include '''Figure X'''. ...
&lt;!-- Cite --&gt;
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
# Consider including quiz question(s) about the take-home messages
&lt;!-- Tables --&gt;
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Very good
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## capitalisation
## [[Help:Wikitext quick reference|italicisation]]
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Very good
## Use internal link style for Wikipedia links (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- External links --&gt;
# External links
## Excellent
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
|9=
&lt;!-- User page --&gt;
# Used effectively
# Brief description about self – consider expanding
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
|10=
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
# The Wikiversity talk page link goes an edit rather than read interface
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:01, 30 August 2026 (UTC)</text>
      <sha1>0c7jy30dltwkpxb5bwhogqq9w3wz6ui</sha1>
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  <page>
    <title>User:U3228742</title>
    <ns>2</ns>
    <id>331364</id>
    <revision>
      <id>2829806</id>
      <parentid>2825609</parentid>
      <timestamp>2026-08-31T01:04:31Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>added headings to page</comment>
      <origin>2829806</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="644" sha1="swvwo1c507oyyqesah48xu2rfc37oh6" xml:space="preserve">== About me ==
I am a student studying an undergraduate Bachelor of Science in Psychology. 

My interests and area of focus is social psychology, emotional intelligence, and the understanding people's motivations. I am a life learner who has studied other health modalities and would love to practice clinical psychology. 

== Hobbies ==
My hobbies are painting, reading and my latest obsession is baking sourdough.

== Chapter I am currently working on ==
Created a chapter on [[Motivation and emotion/Book/2026/Dark empathy|Dark Empathy]] in the [[Motivation and emotion/Book/2026|Motivation and emotion book 2026]]

== Social contibutions ==</text>
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      <text bytes="608" sha1="2nqo56us8jbje58nam3pygsnakeqf8p" xml:space="preserve">== About me ==
I am a student studying an undergraduate Bachelor of Science in Psychology. 

My interests and area of focus is social psychology, emotional intelligence, and the understanding people's motivations. I am a life learner who has studied other health modalities and would love to practice clinical psychology. 

== Hobbies ==

* Painting
* Reading
* Baking sourdough.

== Chapter I am currently working on ==
Created a chapter on [[Motivation and emotion/Book/2026/Dark empathy|Dark Empathy]] in the [[Motivation and emotion/Book/2026|Motivation and emotion book 2026]]

== Social contibutions ==</text>
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      <text bytes="759" sha1="7ot66o57wzhplam5nyblu6b0tu955w8" xml:space="preserve">== About me ==
I am a student studying an undergraduate Bachelor of Science in Psychology. 

My interests and area of focus is social psychology, emotional intelligence, and the understanding people's motivations. I am a life learner who has studied other health modalities and would love to practice clinical psychology. 
[[File:Sourdough Bread Loaf.jpg|thumb|'''Figure 1.''' Sourdough is time consuming process yet worth it for every bite]]

== Hobbies ==

* [[w:Painting|Painting]]
* Reading
* Baking [[w:Sourdough|sourdough]]

== Chapter I am currently working on ==
Created a chapter on [[Motivation and emotion/Book/2026/Dark empathy|Dark Empathy]] in the [[Motivation and emotion/Book/2026|Motivation and emotion book 2026]]

== Social contributions ==</text>
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        <id>3005570</id>
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      <comment>/* Social contributions */  added bullpoints</comment>
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      <text bytes="821" sha1="cg3nokm3ukig5p4uaa0ehwbfejdxcth" xml:space="preserve">== About me ==
I am a student studying an undergraduate Bachelor of Science in Psychology. 

My interests and area of focus is social psychology, emotional intelligence, and the understanding people's motivations. I am a life learner who has studied other health modalities and would love to practice clinical psychology. 
[[File:Sourdough Bread Loaf.jpg|thumb|'''Figure 1.''' Sourdough is time consuming process yet worth it for every bite]]

== Hobbies ==

* [[w:Painting|Painting]]
* Reading
* Baking [[w:Sourdough|sourdough]]

== Chapter I am currently working on ==
Created a chapter on [[Motivation and emotion/Book/2026/Dark empathy|Dark Empathy]] in the [[Motivation and emotion/Book/2026|Motivation and emotion book 2026]]

== Social contributions ==
#Fixed Typo
#Comment on Wikiversity page
#Added to discussion</text>
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      <comment>/* Social contributions */ updated two links</comment>
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      <text bytes="1108" sha1="8otcfiv5hcp9jbpeq672b4ojggwja92" xml:space="preserve">== About me ==
I am a student studying an undergraduate Bachelor of Science in Psychology. 

My interests and area of focus is social psychology, emotional intelligence, and the understanding people's motivations. I am a life learner who has studied other health modalities and would love to practice clinical psychology. 
[[File:Sourdough Bread Loaf.jpg|thumb|'''Figure 1.''' Sourdough is time consuming process yet worth it for every bite]]

== Hobbies ==

* [[w:Painting|Painting]]
* Reading
* Baking [[w:Sourdough|sourdough]]

== Chapter I am currently working on ==
Created a chapter on [[Motivation and emotion/Book/2026/Dark empathy|Dark Empathy]] in the [[Motivation and emotion/Book/2026|Motivation and emotion book 2026]]

== Social contributions ==
#[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FBreathing_exercises_and_relaxation&amp;diff=2829879&amp;oldid=2828782 Fixed '''title''']
#[https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2026/Cognitive_versus_affective_empathy&amp;action=edit Comment on Wikiversity page]
#Added to discussion</text>
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  <page>
    <title>User:WonderfulKitten3</title>
    <ns>2</ns>
    <id>331397</id>
    <revision>
      <id>2829816</id>
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      <timestamp>2026-08-31T02:58:50Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
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      <text bytes="1555" sha1="d7dmvbafca5cd5fy0728s2j4plpf7vg" xml:space="preserve">== Self-Introduction ==
Hello, I am Hung. I currently study a Bachelors of Science in Psychology at the University of Canberra and am on my third year. This account has been made for an assignment assessing topic development from a variety of psychological scientific articles, where I have chosen to undertake &quot;Feedback Literacy&quot; as my focus. 

For the better part of seven or so years, I've actually just worked as an assistant baker for my family business, so I don't actually have much experience '''in''' the field of psychology - however, I am pretty motivated and eager to learn more about how we function as humans so that's better than nothing.

== Socials ==
https://www.instagram.com/hung_.nguyen/

== Social Contributions ==

# [[Motivation and emotion/Book/2025/Uncanny valley and emotion|04:51, 27 August 2026. WonderfulKitten3 explained the abbreviated terms: CGI and AI, in the Motivation and Emotion 2025 book chapter on Uncanny Valley and Emotion.]]
# [[Motivation and emotion/Book/2025/Pleasure anticipation and dopamine|04:57, 27 August 2026. WonderfulKitten3 fixed a grammatical error (changed a period into a colon to indicate a list of items) in the Motivation and Emotion 2025 book chapter on Pleasure Anticipation and Dopamine.]]
# [[Motivation and emotion/Book/2025/Emotion regulation through exercise|04:59, 27 August 2026. WonderfulKitten3 added periods to indicate the end of a sentence and fixed the spelling error (acheive to achieve) in the Motivation and Emotion 2025 book chapter on Emotion Regulation through Exercise.]]</text>
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  <page>
    <title>Managing the good and the evils of social media</title>
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    <id>331420</id>
    <revision>
      <id>2829765</id>
      <parentid>2825331</parentid>
      <timestamp>2026-08-30T19:15:59Z</timestamp>
      <contributor>
        <username>DavidMCEddy</username>
        <id>218607</id>
      </contributor>
      <comment>wdsmth</comment>
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      <text bytes="6565" sha1="2mr94sca75lp9q5n4jy5fb43f69kw9k" xml:space="preserve">:''This discusses a 2026-08-27 interview with [[w:Renée DiResta|Renée DiResta]] about managing the good and the evils of social media, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-08-22 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''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]]&lt;ref name=NPOV&gt;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.&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[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|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;
&lt;!--[[File:Managing the good and the evil of social media.webm|thumb|Interview with Georgetown University Public Policy Professor on managing the good and the evil of social media]]--&gt;
&lt;!--[[File:Managing the good and the evil of social media.ogg|thumb|29:00 mm:ss excerpts from an interview with Georgetown University Public Policy Professor on managing the good and the evil of social media]]--&gt;

[[w:Renée DiResta|Renée DiResta]] discusses what she recommends that individuals and groups, including governments, do to advance their interests considering the good and the evils of social media. Her comments are based primarily on research summarized in her 2024 book on ''Invisible Rulers: The People Who Turn Lies into Reality'',&lt;ref&gt;DiResta (2024).&lt;/ref&gt; updated in 2026 with the addition of an epilogue.&lt;ref&gt;DiResta (2026).&lt;/ref&gt; 

In November 2014, a month before a [[w:Disneyland measles outbreak|major measles outbreak began in Disneyland]], DiResta was the mother of a 12-month-old studying data on vaccination rates in California. She was alarmed that vaccination rates had been falling. She wrote a blog post about that, then called her representative in the California state assembly to ask about the prospects for eliminating the personal-belief exemption from vaccination requirements for children in day care and schools in California. She was told that the anti-vaxxers were too strong. 

After the Disneyland outbreak, she called back and got a different answer: A representative who was an MD had introduced a bill  &quot;to eliminate the personal-belief exemption&quot; to the requirement to vaccinate children entering pre-school and school.  

Vaccines are never 100% effective. Just over half of those infected from the Disneyland outbreak had been vaccinated, proving that 

:''each unvaccinated human threatens everyone else'', 

though the threats to the vaccinated are substantially less than to the unvaccinated.&lt;ref&gt;See also Graves and Samuelson (2022-03).&lt;/ref&gt; 

DiResta publicly supported the bill to eliminate the personal-believe exemption and drew the ire of leading anti-vaxxers, few of whom lived in California. That led to death threats, congressional subpoenas, lawsuits, and legal bills exceeding a million dollars.&lt;ref&gt;DiResta (2024, pp. 1-3).&lt;/ref&gt; She soon figured out that most of the anti-vaccine comments came from out of state, some by bots. 

That work morphed into a job as the Technical Research Manager at the [[w:Stanford Internet Observatory|Stanford Internet Observatory]] between 2019 and 2024 and then to her current position at Georgetown University.&lt;ref&gt;&lt;!--Renee DiResta, Associate Research Professor, Georgetown U--&gt;{{cite Q|Q141158430}}&lt;/ref&gt; That work produced a 2022 report by The Virality Project&lt;ref&gt;&lt;!--The Virality Project--&gt;{{cite Q|Q141158012}}&lt;/ref&gt; with DiResta as lead editor that documents how anti-vaccine narratives on social media dramatically reduced the effectiveness of public health efforts.&lt;ref&gt;The Virality Project (2022).&lt;/ref&gt; It also produced a 2024 report on &quot;Shaping the Future of Social Media with Middleware&quot;, which she edited with Luke Hogg,&lt;ref&gt;Hogg and DiResta, eds. (2024).&lt;/ref&gt; and a 2025 report on &quot;For Expertise to Matter, Nonpartisan Institutions Need New Communications Strategies&quot;, co-authored by Rachel Kleinfeld.&lt;ref&gt;DiResta and Kleinfeld (2025).&lt;/ref&gt;   

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]].

==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]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!--Renée DiResta (2024) Invisible Rulers: The People Who Turn Lies into Reality--&gt;{{cite Q|Q135530439}}
* &lt;!--Renée DiResta (2026) Invisible Rulers: The People Who Turn Lies into Reality, updated edition--&gt;{{cite Q|Q141156783}}
* &lt;!--Renée DiResta (2025-09-10) For Expertise to Matter, Nonpartisan Institutions Need New Communications Strategies--&gt;{{cite Q|Q141158362}}
* &lt;!--Spencer Graves and Douglas A. Samuelson (2022-03) &quot;Externalities, public goods, and infectious diseases--&gt;{{cite Q|Q111367750}}
* &lt;!--Luke Hogg and Renée DiResta, eds. (2024-12) &quot;Shaping the Future of Social Media with Middleware--&gt;{{cite Q|Q141158102}}
*&lt;!--The Virality Project (2022) Memes, Magnets, and Microchips: Narrative dynamics around COVID-19 vaccines--&gt;{{cite Q|Q141158003}}

[[Category:Media]]
[[Category:Social media]]
[[Category:News]]
[[Category:Democracy]]
[[Category:Politics]]
[[Category:Education]]
[[Category:Media literacy]]
[[Category:Media reform to improve democracy]]
&lt;!--list of categories
https://en.wikiversity.org/wiki/Wikiversity:Category_Review
[[Wikiversity:Category Review]]--&gt;</text>
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    <revision>
      <id>2829766</id>
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      <timestamp>2026-08-30T19:18:50Z</timestamp>
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        <username>DavidMCEddy</username>
        <id>218607</id>
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      <comment>wdsmth</comment>
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      <text bytes="6572" sha1="n1npr6szrfggv2cqrfdm7tl6x0j3jvs" xml:space="preserve">:''This discusses a 2026-08-27 interview with [[w:Renée DiResta|Renée DiResta]] about managing the good and the evils of social media, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-08-22 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''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]]&lt;ref name=NPOV&gt;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.&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[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|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;
&lt;!--[[File:Managing the good and the evil of social media.webm|thumb|Interview with Georgetown University Public Policy Professor on managing the good and the evil of social media]]--&gt;
&lt;!--[[File:Managing the good and the evil of social media.ogg|thumb|29:00 mm:ss excerpts from an interview with Georgetown University Public Policy Professor on managing the good and the evil of social media]]--&gt;

[[w:Renée DiResta|Renée DiResta]] discusses what she recommends that individuals and groups, including governments, do to advance their interests considering the good and the evils of social media. Her comments are based primarily on research summarized in her 2024 book on ''Invisible Rulers: The People Who Turn Lies into Reality'',&lt;ref&gt;DiResta (2024).&lt;/ref&gt; updated in 2026 with the addition of an epilogue.&lt;ref&gt;DiResta (2026).&lt;/ref&gt; 

In November 2014, a month before a [[w:Disneyland measles outbreak|major measles outbreak began in Disneyland]], DiResta was the mother of a 12-month-old studying data on vaccination rates in California. She was alarmed that vaccination rates had been falling. She wrote a blog post about that. Then she called her representative in the California state assembly and asked about the prospects for eliminating the personal-belief exemption from vaccination requirements for children in day care and schools in California. She was told that the anti-vaxxers were too strong. 

After the Disneyland outbreak, she called back and got a different answer: A representative who was an MD had introduced a bill  &quot;to eliminate the personal-belief exemption&quot; to the requirement to vaccinate children entering pre-school and school.  

Vaccines are never 100% effective. Just over half of those infected from the Disneyland outbreak had been vaccinated, proving that 

:''each unvaccinated human threatens everyone else'', 

though the threats to the vaccinated are substantially less than to the unvaccinated.&lt;ref&gt;See also Graves and Samuelson (2022-03).&lt;/ref&gt; 

DiResta publicly supported the bill to eliminate the personal-believe exemption and drew the ire of leading anti-vaxxers, few of whom lived in California. That led to death threats, congressional subpoenas, lawsuits, and legal bills exceeding a million dollars.&lt;ref&gt;DiResta (2024, pp. 1-3).&lt;/ref&gt; She soon figured out that most of the anti-vaccine comments came from out of state, some by bots. 

That work morphed into a job as the Technical Research Manager at the [[w:Stanford Internet Observatory|Stanford Internet Observatory]] between 2019 and 2024 and then to her current position at Georgetown University.&lt;ref&gt;&lt;!--Renee DiResta, Associate Research Professor, Georgetown U--&gt;{{cite Q|Q141158430}}&lt;/ref&gt; That work produced a 2022 report by The Virality Project&lt;ref&gt;&lt;!--The Virality Project--&gt;{{cite Q|Q141158012}}&lt;/ref&gt; with DiResta as lead editor that documents how anti-vaccine narratives on social media dramatically reduced the effectiveness of public health efforts.&lt;ref&gt;The Virality Project (2022).&lt;/ref&gt; It also produced a 2024 report on &quot;Shaping the Future of Social Media with Middleware&quot;, which she edited with Luke Hogg,&lt;ref&gt;Hogg and DiResta, eds. (2024).&lt;/ref&gt; and a 2025 report on &quot;For Expertise to Matter, Nonpartisan Institutions Need New Communications Strategies&quot;, co-authored by Rachel Kleinfeld.&lt;ref&gt;DiResta and Kleinfeld (2025).&lt;/ref&gt;   

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]].

==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]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!--Renée DiResta (2024) Invisible Rulers: The People Who Turn Lies into Reality--&gt;{{cite Q|Q135530439}}
* &lt;!--Renée DiResta (2026) Invisible Rulers: The People Who Turn Lies into Reality, updated edition--&gt;{{cite Q|Q141156783}}
* &lt;!--Renée DiResta (2025-09-10) For Expertise to Matter, Nonpartisan Institutions Need New Communications Strategies--&gt;{{cite Q|Q141158362}}
* &lt;!--Spencer Graves and Douglas A. Samuelson (2022-03) &quot;Externalities, public goods, and infectious diseases--&gt;{{cite Q|Q111367750}}
* &lt;!--Luke Hogg and Renée DiResta, eds. (2024-12) &quot;Shaping the Future of Social Media with Middleware--&gt;{{cite Q|Q141158102}}
*&lt;!--The Virality Project (2022) Memes, Magnets, and Microchips: Narrative dynamics around COVID-19 vaccines--&gt;{{cite Q|Q141158003}}

[[Category:Media]]
[[Category:Social media]]
[[Category:News]]
[[Category:Democracy]]
[[Category:Politics]]
[[Category:Education]]
[[Category:Media literacy]]
[[Category:Media reform to improve democracy]]
&lt;!--list of categories
https://en.wikiversity.org/wiki/Wikiversity:Category_Review
[[Wikiversity:Category Review]]--&gt;</text>
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  <page>
    <title>Wikiversity talk:Community Review/Removal of Wikidebates</title>
    <ns>5</ns>
    <id>331424</id>
    <revision>
      <id>2829730</id>
      <parentid>2826214</parentid>
      <timestamp>2026-08-30T14:34:35Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>/* A way to enforce the closure of Wikidebates */ reply to Atcovi ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>2829730</origin>
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      <text bytes="3081" sha1="el0d52vkz6l5tjnt5q8991k86of78ho" xml:space="preserve">== A way to enforce the closure of Wikidebates ==

I can create an abuse filter to warn or prevent users from creating Wikidebate pages. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:46, 23 August 2026 (UTC)

:I have no objections to this. Though, how exactly would we implement such an abuse filter? What are the triggers for such an abuse filter? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:01, 24 August 2026 (UTC)
:: This would detect common interrogative adverbs in the start of the page title, in the main page namespace. I do have the expertise to configure such filters. Again, would it be set to warn or disallow? [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:02, 24 August 2026 (UTC)
:::I would think a warning would be fine. If there is a need to disallow, then we can implement that. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 04:53, 26 August 2026 (UTC)
:::: @[[User:Atcovi|Atcovi]] and @[[User:Jtneill|Jtneill]], I can create the filter, while one of you can create the warning at [[MediaWiki:Abusefilter-warning-wikidebates]]. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 14:34, 30 August 2026 (UTC)
: Has there been a problem with new debates being created since this community review? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:31, 24 August 2026 (UTC)
:: I believe unregistered users might still edit Wikidebates pages even when they were marked as historical, so I can create a quick filter to prevent this since Wikidebates was discontinued. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:57, 24 August 2026 (UTC)
::In fairness, we've had at least a few instances where contributors have tried to add/create new Wikidebates: [[Talk:Does God exist?#Should this proof be added to the Pro list???Talk:Does God exist?#Should this proof be added to the Pro list???|1]], [https://en.wikiversity.org/wiki/Wikiversity:Request_custodian_action#Please_unban 2], [https://en.wikiversity.org/w/index.php?title=Talk:Does_God_exist%3F&amp;diff=prev&amp;oldid=2807096 3] (tho this is an LTA). It isn't a far-fetched idea to implement a technical preventor beyond the notice. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:02, 24 August 2026 (UTC)
: An alternative approach could be to protect pages which attract persistent, problematic editing. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:25, 24 August 2026 (UTC)</text>
      <sha1>el0d52vkz6l5tjnt5q8991k86of78ho</sha1>
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    <revision>
      <id>2829823</id>
      <parentid>2829730</parentid>
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      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>/* A way to enforce the closure of Wikidebates */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>2829823</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3295" sha1="d91zhi00ophui76oxwitds51dqcftlu" xml:space="preserve">== A way to enforce the closure of Wikidebates ==

I can create an abuse filter to warn or prevent users from creating Wikidebate pages. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:46, 23 August 2026 (UTC)

:I have no objections to this. Though, how exactly would we implement such an abuse filter? What are the triggers for such an abuse filter? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:01, 24 August 2026 (UTC)
:: This would detect common interrogative adverbs in the start of the page title, in the main page namespace. I do have the expertise to configure such filters. Again, would it be set to warn or disallow? [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:02, 24 August 2026 (UTC)
:::I would think a warning would be fine. If there is a need to disallow, then we can implement that. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 04:53, 26 August 2026 (UTC)
:::: @[[User:Atcovi|Atcovi]] and @[[User:Jtneill|Jtneill]], I can create the filter, while one of you can create the warning at [[MediaWiki:Abusefilter-warning-wikidebates]]. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 14:34, 30 August 2026 (UTC)
: Has there been a problem with new debates being created since this community review? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:31, 24 August 2026 (UTC)
:: I believe unregistered users might still edit Wikidebates pages even when they were marked as historical, so I can create a quick filter to prevent this since Wikidebates was discontinued. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:57, 24 August 2026 (UTC)
::In fairness, we've had at least a few instances where contributors have tried to add/create new Wikidebates: [[Talk:Does God exist?#Should this proof be added to the Pro list???Talk:Does God exist?#Should this proof be added to the Pro list???|1]], [https://en.wikiversity.org/wiki/Wikiversity:Request_custodian_action#Please_unban 2], [https://en.wikiversity.org/w/index.php?title=Talk:Does_God_exist%3F&amp;diff=prev&amp;oldid=2807096 3] (tho this is an LTA). It isn't a far-fetched idea to implement a technical preventor beyond the notice. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:02, 24 August 2026 (UTC)
: An alternative approach could be to protect pages which attract persistent, problematic editing. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:25, 24 August 2026 (UTC)
: See [[Special:AbuseFilter/23]] and [[Special:AbuseFilter/24]]. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 03:50, 31 August 2026 (UTC)</text>
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  <page>
    <title>Talk:Motivation and emotion/Book/2026/Self-disclosure and emotional intimacy</title>
    <ns>1</ns>
    <id>331432</id>
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      <timestamp>2026-08-31T05:28:19Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Topic development feedback</comment>
      <origin>2829833</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6404" sha1="gsars3ye3j6u5510ujabpdd2uwq8x9g" xml:space="preserve">== Scenario Suggestion - From U3275775 ==

For your overview, I think you could add a scenario about a coffee shop worker and their regular customer who comes in everyday. At first they would only make small talk but as the customer comes in each day, their talks get longer. This would demonstrate how self-disclosure develops gradually over-time, and how this leads to emotional intimacy. [[User:U3275775|U3275775]] ([[User talk:U3275775|discuss]] • [[Special:Contributions/U3275775|contribs]]) 04:56, 24 August 2026 (UTC)

== Resource ==

Hello! I am loving the start to this topic it looks good so far and I'm interested to see how it develops. I found a resource that may be helpful, it's an article about brain hyperscanning to better understand the brain mechanisms that underlie emotional intimacy through self-disclosure. I'll paste the link and I hope it is helpful or at least points you to some extra resorces. Cheers!

https://link.springer.com/article/10.1186/s40359-026-04257-3 [[User:J.M.A Watson|J.M.A Watson]] ([[User talk:J.M.A Watson|discuss]] • [[Special:Contributions/J.M.A Watson|contribs]]) 21:54, 27 August 2026 (UTC)

== Extra Resource ==

Hey, great job so far! I noticed out topics had some similarities and thought you might benefit from this source I've had a look at :) 

Reis, H. T., &amp; Shaver, P. (1988). &lt;nowiki&gt;''&lt;/nowiki&gt;Intimacy as an Interpersonal Process&lt;nowiki&gt;''&lt;/nowiki&gt; (pp. 367–389). Routledge. &lt;nowiki&gt;https://www.researchgate.net/publication/347687013_Intimacy_as_an_interpersonal_process&lt;/nowiki&gt; [[User:U3282586|U3282586]] ([[User talk:U3282586|discuss]] • [[Special:Contributions/U3282586|contribs]]) 10:55, 28 August 2026 (UTC)

== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:U3283302|Hi [[User:U3283302|U3283302]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:28, 31 August 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
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&lt;!-- Title --&gt;
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# Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings
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&lt;!-- Cite --&gt;
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&lt;!-- Learning feature --&gt;
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# Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
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# What are the most relevant systematic reviews/meta-analyses about this topic?
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# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## use dois where available instead of other links
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# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
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# Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:28, 31 August 2026 (UTC)</text>
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  <page>
    <title>Motivation and emotion/Book/2026/Feedback literacy</title>
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        <username>Jtneill</username>
        <id>10242</id>
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      <comment>Copyediting</comment>
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      <text bytes="9920" sha1="71t0qi350d2q08zgwubcqhm3iaibbhn" xml:space="preserve">{{title|Feedback literacy:&lt;br&gt;What is feedback literacy, why is it important, and how can it be developed?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=2}}

[[File:Young Man Pondering - Tangier - Morocco.jpg|right|thumb|200px|'''Figure 1'''. Self-reflection is one of humanities{{g}} greatest powers {{ic|Explain how clearly how this image connects with the text}}]]

; Food for thought
Who remembers being told that they possess potential but are too unserious to step up and wield it?

Maybe it wasn't a universal experience and just mine{{who}}, but the point remains; knowledge is more effectively learned if the individual is actively engaged with it, rather than being a passive participant to potential experiences (see Figure 1).
{{RoundBoxBottom}}

Feedback literacy refers to the knowledge and skills that enable individuals to understand, evaluate, and utilise feedback to improve their current and future performance and is important as it promotes learning agency/independence and emotional self-regulation through developing evaluative judgement capacity (Carless &amp; Boud, 2018; Molloy et al., 2020). This benefits not only academic performance but is transferrable to future professional and future life-skills development (Tai et al., 2018; Winstone et al., 2017). 

Feedback literacy can be developed through purposeful curriculum design. Effective strategies include analysing exemplars, engaging in peer-reviews, incorporating feedback dialogues (feedback '''during''' the drafting phase), and self-revision (Boud &amp; Molloy, 2013). {{RoundBoxTop|theme=2}}
'''Focus questions'''

* What is Feedback Literacy?
* Why Does it Matter?
* How can it be Developed?

{{RoundBoxBottom}}

==What is Feedback Literacy?==

Feedback literacy is an integral component of broader academic literacy and refers to the knowledge, attitudes, and skills that learners possess to enable effective evaluation and understanding of feedback, enhancing learning and future performance. Feedback literacy is developed both individually and rationally through peer, teacher, assessment interactions, and disciplinary standards; it involves '''actively engaging with feedback, judging relevance, managing emotional affect, and applying it practically''' (Carless &amp; Boud, 2018; Sutton, 2012).

=== Appreciating Feedback ===
Feedback-literate individuals recognise that feedback is not just an explanation of grades, a correction of mistakes, or a judgement of ability, but rather as information that identifies strengths and weaknesses in ability, academic and/or professional standards, and the gap between current and desired performance (Carless &amp; Boud, 2018). Critical feedback is more effective if students perceive it from a dialogue perspective and should emphasise the purpose, context, and source of feedback rather than being passive recipients to what could be considered harsh criticism (Sutton, 2012). For example, feedback-literate students may ask: ''what is this feedback helping me learn? Which assessment criterion does this relate to? How might I apply this advice to another task?''

=== Making Judgements ===
Making judgements is the ability to determine:

* what constitutes as high-quality work
* how well an individual's work meets the relevant criteria
* what specific improvements are required (Carless &amp; Boud, 2018).

This is intimately associated with evaluative judgement, which is the capacity to make educated decisions about an individual and other's quality of work (Tai et al., 2017). Academic judgement is dynamic in evaluation as competent students should consider whether feedback is accurate, relevant and consistent to the assessment criteria, and applicable to their learning goals (Boud &amp; Molloy, 2012).

=== Managing Affect ===
Managing emotional affect from criticism is a vital capacity in feedback-literacy. Emotions such as: anxiety, embarrassment, disappointment, defensiveness, and/or frustration are common emotions involved with receiving criticism; remaining conscious and regulating these emotions is key to engaging with useful feedback (Winstone et al., 2017; Carless &amp; Boud, 2018). Crucially, individuals must remove self-identity and worth from criticism to their work, as inability to manage emotions often lead to &quot;avoidance strategies&quot; such as ignoring feedback entirely (Winstone et al., 2017).

{{RoundBoxTop|theme=2}}
;Did you know?
A case study conducted by Baron (1988) studied the effects of destructive and constructive criticism and found that destructive criticism lead to greater anger and tension, indicating that future disagreements were likely to be handled with resistance and/or avoidance rather than collaboration and compromise.
{{RoundBoxBottom}}

=== Taking Action ===
Often referred to as &quot;uptake&quot;, taking action is the process of translating feedback into observable improvements in performance (Carless &amp; Boud, 2018). The hardest part of uptake is the &quot;implementation gap&quot; - the difficulty of knowing how to act on specific comments, as taking action requires the ability to convert general advice into actionable and specific strategies (Winstone &amp; Carless, 2019; Boud &amp; Molloy, 2013). Highly adept individuals can translate specific criticism into broader unrelated tasks, thereby demonstrating &quot;transfer&quot; of knowledge (Molloy et al., 2019).

== Why is Feedback Literacy Important? ==
Feedback literacy is a core academic competency as it allows individuals to transform feedback from a transactional interaction into a self-sustainable learning process. Aside from facilitating for self regulated learning, feedback literacy develops fundamental understanding in '''evaluative judgement''', mitigates the '''&quot;feedback gap&quot;''', improves '''emotional regulation and academic wellbeing''', and '''supports professional learning''' - many of which have been identified in the previous section, therefore professional learning is emphasised here.

Feedback literacy is a pillar of professional identity through the constant negotiation between self-concept and external appraisals. Advanced feedback-literate individuals can differentiate between criticism that is targeted to their worth or work; mature psychological separation is crucial to maintaining long-term motivation and avoiding burnout (Molloy et al., 2019).

== How can Feedback Literacy be Developed? ==
Feedback literacy can be pedagogically developed through '''peer review, use of exemplars, feedback dialogues,''' and '''meta-cognitive reflection''', all requiring structured opportunities so that individuals can actively engage with feedback and calibrate judgement processes.

==Conclusion==
Feedback literacy is a vital learning competency that enables individuals to interpret, evaluate, and transform transactional feedback into self-cultivating improvements; learners become lifelong active participants of their feedback through developing evaluative judgement and managing emotional affects to take effective and intentional action. Strategies such as peer-review, use of exemplars, feedback dialogues, and self-reflection strengthen student independence and self-regulation, ultimately leading to an ongoing process supporting continuous improvement.

==See also==
* [[Motivation and emotion/Book/2026/Expectancy-value theory of educational motivation|Expectancy-value theory of educational motivation]] (Book chapter, 2026)
* [[Motivation and emotion/Book/2026/Extended process model of emotion regulation|Extended process model of emotion regulation]] (Book chapter, 2026)
* [[wikipedia:Emotional_self-regulation|Emotional self-regulation]] (Wikipedia)

==References==
{{Hanging indent|1=

Baron, R. A. (1998). Negative effects of destructive criticism: Impact on conflict, self-efficacy, and task performance. ''Journal of Applied Psychology, 73''(2), 199–207. https://doi.org/10.1037/0021-9010.73.2.199

Boud, D., &amp; Molloy, E. (2013). Rethinking models of feedback for learning: the challenge of design. ''Assessment &amp; Evaluation in Higher Education, 38''(6), 698–712. https://doi.org/10.1080/02602938.2012.691462

Carless, D., &amp; Boud, D. (2018). The development of student feedback literacy: enabling uptake of feedback. ''Assessment &amp; Evaluation in Higher Education, 43''(8), 1315–1325. https://doi.org/10.1080/02602938.2018.1463354

Molloy, E., Boud, D., &amp; Henderson, M. (2020). Developing a learning-centred framework for feedback literacy.'' Assessment &amp; Evaluation in Higher Education, 45''(4), 527–540. https://doi.org/10.1080/02602938.2019.1667955

Sutton, P. (2012). Conceptualizing feedback literacy: knowing, being, and acting. ''Innovations in Education and Teaching International, 49''(1), 31–40. https://doi.org/10.1080/14703297.2012.647781

Tai, J., Ajjawi, R., Boud, D., Dawson, P., &amp; Panadero, E. (2018). Developing evaluative judgement: Enabling students to make decisions about the quality of work. ''Higher Education, 76''(3), 467–481. https://link.springer.com/article/10.1007/s10734-017-0220-3

Winstone, N. E., &amp; Carless, D. (2019). Designing effective feedback processes in higher education: A learning-focused approach. ''Routledge''. https://doi.org/10.4324/9781351115940

Winstone, N. E., Nash, R. A., Parker, M., &amp; Rowntree, J. (2017). Supporting Learners’ Agentic Engagement With Feedback: A Systematic Review and a Taxonomy of Recipience Processes. ''Educational Psychologist, 52''(1), 17–37. https://doi.org/10.1080/00461520.2016.1207538

}}

==External links==
* [https://www.centerforengagedlearning.org/assessing-student-feedback-literacy/ Assessing Student Feedback Literacy] (centreforengagedlearning.org)
* [https://www.gla.ac.uk/myglasgow/learningandteaching/afresourceshub/iterative/feedbackliteracy/ Feedback Literacy] (University of Glasgow) 
* [https://www.feedbackliteracy.org/ Feedback literacy] (feebackliteracy.org)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
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    <title>Motivation and emotion/Book/2026/Game of dice task and decision-making</title>
    <ns>0</ns>
    <id>331455</id>
    <revision>
      <id>2829848</id>
      <parentid>2827677</parentid>
      <timestamp>2026-08-31T08:46:17Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Copyediting</comment>
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      <text bytes="16293" sha1="t2dke1jza5uxelwv36ov7ugrgfpfr4w" xml:space="preserve">{{title|Game of dice task and decision-making:&lt;br&gt;What does the game of dice task reveal about risk-based decision-making?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=2}}
[[File:Gaming dice multisided blue.jpg|right|thumb|200px|'''Figure 1'''. Dice have clearly defined odds {{ic|Explain how this connects to the scenario and topic}}]]

; Imagine this ...

Emile has money to spend on a night out and decides to go to a casino. He arrives at a dice table, where a card in front of him lists his betting options. He could bet on the dealer rolling one specific number for a large payout, but the chances of it happening are low: exactly 1 in 6. 

He could also bet on four numbers at once, winning if the dealer rolls one of them. His payout shrinks significantly, but his odds of winning increase to 2 in 3. These odds are not hidden from Emile; he knows exactly what his chances of winning and losing are (see Figure 1). 

For the first few rolls, Emile spends his money on safer bets. After a couple of small wins, he's feeling confident and switches to the single number bet, hoping for that big win. He loses. He goes for the single number again. He loses again. By the end of the night, despite knowing the odds the whole time, Emile has lost most of his money on the riskiest, least likely bets. 

Why did Emile make riskier choices, even though his chances of winning were so much lower?
{{RoundBoxBottom}}

* The Game of Dice Task (GDT) is a valuable tool for investigating performance under explicit risk. Participants have defined probabilities and consequences. This helps assess the neurological and individual motivations for risk taking (Brand et al., 2005).
* Decision-making depends on cognitive mechanisms as well as individual perceptions of risk. Game of Dice Task (GDT) research implicates probability processing, executive functioning, reasoning, the effects of feedback, individual strategy and memory - in general, executive control (Pertl et al., 2017; Schiebener et al., 2011; Schiebener et al., 2014; Schiebener &amp; Brand, 2015).
* Risk-based decision-making requires people to consider probability, outcomes and potential rewards. Greater rewards with lower chances may or may not be perceived as advantageous: there is a difference between taking an uncertain choice and a risk (Brand et al., 2006; Mishra, 2014).

{{RoundBoxTop|theme=4}}
'''Focus questions'''

* What is the Game of Dice Task?
* What is risk-based decision-making? 
* Which psychological processes affect decision-making under explicit risk?
* What can the Game of Dice Task tell us about developmental and neurological disorders?
* How can the Game of Dice Task be applied to real-world decision-making?

{{RoundBoxBottom}}

== Risk-based decision-making ==

* Risk-based decision-making refers to choices made under 'risk': this means that the outcome of a choice is unknown, but the probability of a given outcome can be known. E.g, rolling a six-sided die has an unknown result, but the probability that any given number will be face-up is 1/6. 
* This is different to uncertainty in which both the outcome ''and'' the probability of that outcome are unknown. The GDT is a measure of risk choices defined in this way. Other measures, like the IGT, combine uncertainty and risk, creating ambiguous risk scenarios (Brand et al., 2006; De Groot &amp; Thurik, 2018).
* People do not make the same choices when presented with the same risk scenarios. This suggests individual differences in perception, risk preference, loss aversion, mathematical processing, affective responses and executive functioning (Lowenstein et al., 2001; Mishra, 2014; Pachur &amp; Zilker, 2026).
* Taking risks is not always an indication of poor decision making. Choices may be more or less advantageous based on reward and probability. 

== The Game of Dice Task ==

* Developed by Brand et al. (2004, 2005) in studying patients with Parkinson's disease and Korsakoff. Performance was significantly impaired in these patients, with specific executive functions and brain regions implicated: the limbic-orbitofrontal-striatal loop and dorsolateral prefrontal-striatal loop
* Participants attempt to maximise money over 18 trials. A single six-sided die is randomly &quot;thrown&quot; (virtually) each trial. Participants bet on the rolled number by selecting pre-determined combinations of 1, 2, 3 or 4 numbers. Selecting one number gives a 1/6 chances of winning, with the highest possible gain, selecting two numbers gives a 1/3 chance, with a slightly lower gain, and so on.
{{font color|red|Author's note: Insert table depicting choices, probability and rewards, as well as figure depicting GDT display.}} 
* Participants receive information about the outcome and current monetary balance, creating a feedback opportunity to monitor success/failure in previous decisions (Brand, 2008).
* Different from previous gambling tasks, e.g the Iowa Gambling Task, in which participants discover strategy through previous experience. Known probabilities distinguish risk-based decision-making under explicit risk vs. uncertainty or learned risk (Brand et al., 2006)

== Psychological mechanisms of risk-based decision-making ==

* The GDT can examine how risk-based decision-making develops, matures, or becomes impaired. Evidence from clinical populations suggests relationships between GDT performance and particular cognitive or neurological systems. Risk-based decision-making changes substantially during childhood and adolescence. Schiebener et al. (2014) examined participants aged 8–19 years and found developmental improvements in GDT performance: likely refined executive control. 
* Brand et al. (2004) found that people with Parkinson's disease performed worse on the GDT than healthy controls, and disadvantageous choices were associated with executive functions and feedback processing. A  review of decision-making under risk and ambiguity in Parkinson's disease concluded that executive functions are impacted, but may play different roles depending on the decision-making task and its cognitive demands (Colautti et al., 2023), further suggesting separate cognitive processes for risk and ambiguity. Other neurological conditions show similar links between executive functioning and GDT performance. For example, people with Korsakoff syndrome show impaired performance on the GDT, with the impairment associated with executive functions (Brand et al., 2005; Brand et al., 2009).
* Feedback processing is also important to GDT performance. Removing feedback (i.e, win or loss, total wins or losses, current and previous balance, etc.) worsened performance in healthy individuals, suggesting cognitive and affective processing are both important for advantageous decision making (Brand, 2008). Amnesic patients showed no difference in GDT performance with or without feedback, consistently performing poorly, illustrating the importance of feedback and experience in avoiding disadvantageous choices (Brand et al., 2009).

{{RoundBoxTop|theme=7}}
&lt;quiz display=simple&gt;
{Executive functioning can influence whether someone makes advantageous choices in the GDT:
|type=&quot;()&quot;}
+ True
- False

{Feedback is irrelevant to GDT performance, since the probabilities are already known:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;
{{RoundBoxBottom}}

== Applications and limitations of the GDT ==

* GDT findings should be interpreted as evidence about a particular form of risk-based decision-making. Different behavioural tasks capture different aspects of decision-making. Laboratory measures of risk and decision-making do not necessarily measure a single unified construct (Buelow &amp; Blaine, 2015; Colautti et al., 2023).
* The controlled nature and specific probabilities of the GDT allows researchers to investigate the contribution of executive control, reasoning, numerical processing and feedback. It has been useful in identifying deficits in cognitive ability amongst clinical populations. This also has some conceptual relevance for real world decisions involving defined risk.
* Many real word scenarios are confounded with uncertainty, social pressures, competing goals and subjective reward considerations. The GDT is not necessarily effective at predicting real-life decision-making (Buelow et al., 2024). 

==Conclusion==
* The GDT provides a controlled way to study decision-making under risk. Because the probabilities and possible outcomes are known, the GDT allows researchers to examine how people use information about risk when making repeated decisions.
* Advantageous decision-making is not simply about avoiding risk. It involves considering the probability and potential outcomes of different choices and using reasoning, executive functioning and feedback to guide decisions. A risky choice is not necessarily a poor decision if the potential reward and probability of success make it advantageous.
* GDT performance is influenced by a range of cognitive processes and individual differences. The GDT therefore helps researchers examine why some people are better able than others to adjust their decisions to the risks and outcomes involved. However, it is not necessarily an appropriate analogue for decision-making in complex real life situations. 

* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[wikipedia:Decision-making|Decision-making]] (Wikipedia)
* [[Motivation and emotion/Book/2024/Dopamine and decision making|Dopamine and decision making]] (Book chapter, 2024)
* [[Motivation and emotion/Book/2018/Gambling addiction motivation|Gambling addiction motivation]] (Book chapter, 2018)
* [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018)
* [[wikipedia:Risk|Risk]] (Wikipedia)

==References==
{{Hanging indent|1=

Brand, M., Fujiwara, E., Borsutzky, S., Kalbe, E., Kessler, J., &amp; Markowitsch, H. J. (2005). Decision-making deficits of korsakoff patients in a new gambling task with explicit rules: Associations with executive functions. ''Neuropsychology'', ''19''(3), 267–277. https://doi.org/10.1037/0894-4105.19.3.267

Brand, M., Labudda, K., Kalbe, E., Hilker, R., Emmans, D., Fuchs, G., Kessler, J., &amp; Markowitsch, H. J. (2004). Decision‐making impairments in patients with Parkinson’s disease. ''Behavioural Neurology'', ''15''(3–4), 77–85. https://doi.org/10.1155/2004/578354

Brand, M., Labudda, K., &amp; Markowitsch, H. J. (2006). Neuropsychological correlates of decision-making in ambiguous and risky situations. ''Neural Networks'', ''19''(8), 1266–1276. https://doi.org/10.1016/j.neunet.2006.03.001

Brand, M., Pawlikowski, M., Labudda, K., Laier, C., Rothkirch, N. von, &amp; Markowitsch, H. J. (2009). Do amnesic patients with Korsakoff’s syndrome use feedback when making decisions under risky conditions? An experimental investigation with the game of dice task with and without feedback. ''Brain and Cognition'', ''69''(2), 279–290. https://doi.org/10.1016/j.bandc.2008.08.003

Brand, Matthias. (2008). Does the feedback from previous trials influence current decisions? A study on the role of feedback processing in making decisions under explicit risk conditions. ''Journal of Neuropsychology'', ''2''(2), 431–443. https://doi.org/10.1348/174866407x220607

Buelow, M. T., &amp; Blaine, A. L. (2015). The assessment of risky decision making: A factor analysis of performance on the Iowa Gambling Task, Balloon Analogue Risk Task, and Columbia Card Task. ''Psychological Assessment'', ''27''(3), 777–785. https://doi.org/10.1037/a0038622

Buelow, M. T., Okdie, B. M., &amp; Kowalsky, J. M. (2024). Ecological validity of common behavioral decision making tasks: Evidence across two samples. ''Neuropsychology, Development, and Cognition. Section A, Journal of Clinical and Experimental Neuropsychology/Journal of Clinical and Experimental Neuropsychology'', ''46''(3), 1–20. https://doi.org/10.1080/13803395.2024.2337759

Colautti, L., Iannello, P., Maria Caterina Silveri, &amp; Antonietti, A. (2023). Decision-making under ambiguity and risk and executive functions in parkinson’s disease patients: A scoping review of the studies investigating the Iowa gambling task and the game of dice. ''Cognitive, Affective, &amp; Behavioral Neuroscience'', ''23''(5), 1225–1243. https://doi.org/10.3758/s13415-023-01106-3

De Groot, K., &amp; Thurik, R. (2018). Disentangling risk and uncertainty: When risk-taking measures are not about risk. ''Frontiers in Psychology'', ''9''. https://doi.org/10.3389/fpsyg.2018.02194

Loewenstein, G. F., Weber, E. U., Hsee, C. K., &amp; Welch, N. (2001). Risk as feelings. ''Psychological Bulletin'', ''127''(2), 267–286. https://doi.org/10.1037/0033-2909.127.2.267

Mishra, S. (2014). Decision-making under risk: Integrating perspectives from biology, economics, and psychology. ''Personality and Social Psychology Review'', ''18''(3), 280–307. https://doi.org/10.1177/1088868314530517

Pachur, T., &amp; Zilker, V. (2026). Psychological theories of decision-making under risk. ''The Cambridge Handbook of Behavioural Data Science'', 76–105. https://doi.org/10.1017/9781108939010.009

Pertl, M.-T., Zamarian, L., &amp; Delazer, M. (2017). Reasoning and mathematical skills contribute to normatively superior decision making under risk: Evidence from the game of dice task. ''Cognitive Processing'', ''18''(3), 249–260. https://doi.org/10.1007/s10339-017-0813-x

Schiebener, J., &amp; Brand, M. (2015). Self-reported strategies in decisions under risk: Role of feedback, reasoning abilities, executive functions, short-term-memory, and working memory. ''Cognitive Processing'', ''16''(4), 401–416. https://doi.org/10.1007/s10339-015-0665-1

Schiebener, J., Wegmann, E., Gathmann, B., Laier, C., Pawlikowski, M., &amp; Brand, M. (2014). Among three different executive functions, general executive control ability is a key predictor of decision making under objective risk. ''Frontiers in Psychology'', ''5''. https://doi.org/10.3389/fpsyg.2014.01386

Schiebener, J., Zamarian, L., Delazer, M., &amp; Brand, M. (2011). Executive functions, categorization of probabilities, and learning from feedback: What does really matter for decision making under explicit risk conditions?. ''Journal of Clinical and Experimental Neuropsychology'', ''33''(9), 1025–1039. https://doi.org/10.1080/13803395.2011.595702 
}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.psychologytoday.com/au/blog/hovercraft-full-eels/202012/risky-gamble-or-uncertain-future A risky gamble or an uncertain future?] (Psychology Today)
* [https://www.millisecond.com/library/gameofdicetask Game of dice task] (Millisecond Software)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Decision making]]</text>
      <sha1>t2dke1jza5uxelwv36ov7ugrgfpfr4w</sha1>
    </revision>
  </page>
  <page>
    <title>Talk:Motivation and emotion/Book/2026/Feedback literacy</title>
    <ns>1</ns>
    <id>331466</id>
    <revision>
      <id>2829818</id>
      <parentid>2828277</parentid>
      <timestamp>2026-08-31T03:13:01Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Topic development feedback</comment>
      <origin>2829818</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6053" sha1="1hops44aoqgw28sxk04whlnqs9dgkx7" xml:space="preserve">== Reference ==

Hi. I know you are just starting this page, so you might find this reference, which is a critical review, helpful in assembling some thoughts. good luck.

Nieminen, J.H., Carless, D. Feedback literacy: a critical review of an emerging concept. ''High Educ'' '''85''', 1381–1400 (2023). &lt;nowiki&gt;https://doi.org/10.1007/s10734-022-00895-9&lt;/nowiki&gt; [[User:StretchBeyond|StretchBeyond]] ([[User talk:StretchBeyond|discuss]] • [[Special:Contributions/StretchBeyond|contribs]]) 22:29, 24 August 2026 (UTC)

== Great chapter structure! ==

&lt;nowiki&gt;Hi! Your chapter is really easy to read, and I love how you broke down the &quot;Taking Action&quot; section. Since this is for the Motivation and Emotion unit, maybe you could add a brief note about how a student's motivation (like wanting a good grade) affects whether they actually use the feedback. Great start! ~~~~&lt;/nowiki&gt; [[User:U3239431|U3239431]] ([[User talk:U3239431|discuss]] • [[Special:Contributions/U3239431|contribs]]) 13:33, 27 August 2026 (UTC)

== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:WonderfulKitten3|Hi [[User:WonderfulKitten3|WonderfulKitten3]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:12, 31 August 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed)
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
# There is a strong tendency to overcapitalise throughout
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# Consider using subheadings in the other main sections
&lt;!-- Alignment with focus questions --&gt;
# Clear alignment between sub-title, focus questions, and heading structure
|3=
&lt;!-- Overview--&gt;
# Scenario, image, evocative description of the problem/topic, and focus questions
# Explain who is using first person in the scenario
# Keep the description user-friendly; consider moving the bulk of the citations into subsequent sections
&lt;!-- Focus questions --&gt;
# Focus questions are aligned with sub-title and top-level headings
# Remove overcapitalisation (use sentence casing)
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Promising development
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# Select the best theories about this topic
# Select the best research about this topic
# Connect to motivation
&lt;!-- GenAI ---&gt;
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is underway
# Connect to motivation
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# The relevance of the figure to the topic is unclear
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Weak use of scenario; consider using a more real-world example
&lt;!-- Quiz --&gt;
# Consider including quiz question(s) about the take-home messages
&lt;!-- Tables --&gt;
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Very good
&lt;!-- Systematic reviews --&gt;
# At least one relevant systematic review and/or meta-analysis has been identified
&lt;!-- Citations --&gt;
# Only include references which have been accessed and read
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## capitalisation
## use dois where available instead of other links
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
## Use alphabetical order
&lt;!-- External links --&gt;
# External links
## Excellent
|9=
&lt;!-- User page --&gt;
# Good
# Description about self provided
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# Add link to book chapter
|10=
&lt;!-- Social contribution --&gt;
# None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]].
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:12, 31 August 2026 (UTC)</text>
      <sha1>1hops44aoqgw28sxk04whlnqs9dgkx7</sha1>
    </revision>
  </page>
  <page>
    <title>Talk:Motivation and emotion/Book/2026/Value congruence and motivation</title>
    <ns>1</ns>
    <id>331468</id>
    <revision>
      <id>2829803</id>
      <parentid>2825379</parentid>
      <timestamp>2026-08-31T00:49:21Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Topic development feedback</comment>
      <origin>2829803</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6443" sha1="ry7em2p3ka3jyubqilbpx4i4wxrt7qg" xml:space="preserve">== Perceived control and value misalignment ==

Hi

I really like your opening scenario with Jane; it makes the topic relatable right away and ties in well with your focus questions.

A couple of thoughts on your outline:

You've got five theories listed SDT, Self-Concordance, Motivational Congruence, P-O Fit, and Amity Goal Orientation. That's a lot to fit into one chapter might be worth picking one as your main theory and using the others more briefly to support it, so it doesn't end up feeling like five separate mini-sections.
I haven't heard of Amity Goal Orientation Theory before, check that there's enough solid research behind it; otherwise, it might work better as a short mention rather than its own section.
For the BPD example under individual differences, it might help to note that this is a different, clinical situation, so readers don't assume it works the same way for everyone.
The perceived vs actual congruence point looks really promising. I think that's actually the key to your original discussion question maybe it's not misalignment itself that hurts motivation, but feeling misaligned.

Great structure overall, keen to see it filled in! [[User:U3239236|U3239236]] ([[User talk:U3239236|discuss]] • [[Special:Contributions/U3239236|contribs]]) 00:47, 25 August 2026 (UTC)

== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:U3275775|Hi [[User:U3275775|U3275775]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:49, 31 August 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
# User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.)
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure

&lt;!-- GenAI ---&gt;
# Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
# It is unclear how the scenario relates to value congruence
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
&lt;!-- Style --&gt;
# Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios)
&lt;!-- Focus questions --&gt;
# Focus questions are aligned with sub-title and top-level headings
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Kkey points are well developed for each section
# My key suggestion is strive to integrate key ideas across several sources, rather than summarising each source in a separate paragraph
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is about very good, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title
&lt;!-- Theory and research --&gt;
# Good balance of theory and research
&lt;!-- Citations --&gt;
# Very good use of citations
# Remove underline (not APA style)

&lt;!-- Conclusion --&gt;
# Conclusion is underway
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Relevant figure(s) presented, captioned, and cited
&lt;!-- Cite --&gt;
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Excellent use of scenarios/examples/case studies
&lt;!-- Quiz --&gt;
# Promising use of quiz question(s)
&lt;!-- Tables --&gt;
# Promising use of table(s)
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Excellent
# Very good
# Good
# Basic
# Insufficient
# To be developed
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## [[Help:Wikitext quick reference|italicisation]]
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
&lt;!-- External links --&gt;
# External links
## Excellent
|9=
&lt;!-- User page --&gt;
# Used effectively
&lt;!-- Description about self --&gt;
# Description about self provided
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
|10=
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
# Well done on uploading images. Thank-you.
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:49, 31 August 2026 (UTC)</text>
      <sha1>ry7em2p3ka3jyubqilbpx4i4wxrt7qg</sha1>
    </revision>
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  <page>
    <title>User:U3261207</title>
    <ns>2</ns>
    <id>331535</id>
    <revision>
      <id>2829820</id>
      <parentid>2827595</parentid>
      <timestamp>2026-08-31T03:15:35Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Social contributions */ {{ic|Use a numbered list as shown in Tutorial 2}}</comment>
      <origin>2829820</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1447" sha1="p04b8p3yzatg6io71k1gha1e3zxnfnr" xml:space="preserve">===== About me =====
My name is Charli. I am currently completing a Bachelor of Science in Psychology at the University of Canberra.

===== Book chapter I am currently working on =====
The Retirement Motivation book chapter for the Motivation and Emotion book (https://w.wiki/Toxx). I am passionate about this chapter because my mother is currently going through this decision; the scenario is based on her real experience so far.

== Social contributions ==
{{ic|Use a numbered list as shown in Tutorial 2}}

1. Edited Pregnancy Loss and Emotion 2025 chapter (https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Pregnancy_loss_and_emotion&amp;oldid=2825414), (https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Pregnancy_loss_and_emotion&amp;oldid=2825415) 

2. Commented on the Pregnancy Loss and Emotion 2025 chapter talk page ([[Talk:Motivation and emotion/Book/2025/Pregnancy loss and emotion#c-U3261207-20260825014200-Topic Development Feedback 2|https://en.wikiversity.org/wiki/Talk:Motivation_and_emotion/Book/2025/Pregnancy_loss_and_emotion#c-U3261207-20260825014200-Topic_Development_Feedback_2]])

3. Discussion post discussing uploading images that have been generated by AI (https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456861?entry_id=808437). Discussion post discussing hyperlink (https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455468?entry_id=810003)</text>
      <sha1>p04b8p3yzatg6io71k1gha1e3zxnfnr</sha1>
    </revision>
    <revision>
      <id>2829821</id>
      <parentid>2829820</parentid>
      <timestamp>2026-08-31T03:16:18Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Book chapter I am currently working on */ {{ic|Use an internal link as shown in Tutorial 2}}</comment>
      <origin>2829821</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1493" sha1="6g58hxfaphr5vdvripe3jgezrj2m2nc" xml:space="preserve">===== About me =====
My name is Charli. I am currently completing a Bachelor of Science in Psychology at the University of Canberra.

== Book chapter I am currently working on ==
{{ic|Use an internal link as shown in Tutorial 2}}

The Retirement Motivation book chapter for the Motivation and Emotion book (https://w.wiki/Toxx). I am passionate about this chapter because my mother is currently going through this decision; the scenario is based on her real experience so far.

== Social contributions ==
{{ic|Use a numbered list as shown in Tutorial 2}}

1. Edited Pregnancy Loss and Emotion 2025 chapter (https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Pregnancy_loss_and_emotion&amp;oldid=2825414), (https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Pregnancy_loss_and_emotion&amp;oldid=2825415) 

2. Commented on the Pregnancy Loss and Emotion 2025 chapter talk page ([[Talk:Motivation and emotion/Book/2025/Pregnancy loss and emotion#c-U3261207-20260825014200-Topic Development Feedback 2|https://en.wikiversity.org/wiki/Talk:Motivation_and_emotion/Book/2025/Pregnancy_loss_and_emotion#c-U3261207-20260825014200-Topic_Development_Feedback_2]])

3. Discussion post discussing uploading images that have been generated by AI (https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456861?entry_id=808437). Discussion post discussing hyperlink (https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455468?entry_id=810003)</text>
      <sha1>6g58hxfaphr5vdvripe3jgezrj2m2nc</sha1>
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  <page>
    <title>Talk:Motivation and emotion/Book/2026/Game of dice task and decision-making</title>
    <ns>1</ns>
    <id>331559</id>
    <revision>
      <id>2829845</id>
      <parentid>2826791</parentid>
      <timestamp>2026-08-31T08:45:03Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Topic development feedback</comment>
      <origin>2829845</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4048" sha1="hpsbwn4x85a8s4gi9fz667n0sz5r62s" xml:space="preserve">== Feedback and Suggestions  ==

Hello! 

I really enjoyed reading what you have so far. I think the opening scenario makes the Game of Dice Task easy to understand and gives a clear example as to why people may take risky decisions even when the odds were obvious. I think it would be interesting to add a learning feature such as a questionnaire that asks the reader what decision they'd make in a similar situation as your scenario described. I think it would make the page more interactive and allow the reader to have a GDT experience of their own. Just a suggestion! 

Thanks!

- Amy :) [[User:Amyuniversity|Amyuniversity]] ([[User talk:Amyuniversity|discuss]] • [[Special:Contributions/Amyuniversity|contribs]]) 16:33, 26 August 2026 (UTC)

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
# Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand)
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
&lt;!-- Alignment with focus questions --&gt;
# Reasonably good alignment between focus questions and heading structure, but aim for closer alignment
|3=
&lt;!-- Overview--&gt;
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
&lt;!-- Description --&gt;
# A promising description of the problem/topic is planned or presented
&lt;!-- Focus questions --&gt;
# Develop closer alignment between the sub-title, focus questions, and top-level headings
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Promising development
# Connect to motivation
# Highlight the most relevant theories and synthesise the best research on the topic
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is underway
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Relevant figure(s) presented, captioned, and cited
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
# Promising use of quiz question(s)
&lt;!-- Tables --&gt;
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Excellent
&lt;!-- Systematic reviews --&gt;
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- Citations --&gt;
# Only include references which have been accessed and read
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
&lt;!-- External links --&gt;
# External links
## Excellent
|9=
&lt;!-- User page --&gt;
# Used effectively
&lt;!-- Description about self --&gt;
# Description about self provided
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
|10=
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:44, 31 August 2026 (UTC)</text>
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  </page>
  <page>
    <title>User:U3280743</title>
    <ns>2</ns>
    <id>331564</id>
    <revision>
      <id>2829841</id>
      <parentid>2827558</parentid>
      <timestamp>2026-08-31T07:22:17Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Social Contributions */ {{ic|Use a numbered list as shown in Tutorial 2}}</comment>
      <origin>2829841</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1569" sha1="b1g1nikxx2c7jcrsxk6fq2ze9b5o9gp" xml:space="preserve">== About Me ==
Hi! My name is Isabella. I'm a third year Psychology / Politics and International Relations student at the [https://www.canberra.edu.au/?utm_source=google&amp;utm_medium=search&amp;utm_campaign=Always+On+Domestic+Search+2026&amp;utm_term=exact+match&amp;utm_content=study+at+uc&amp;ef_id=Cj0KCQjwnbrUBhDOARIsAKKhPpc52C3Utu5KibqLODSYHjrKrW0jG7_rQP9ghGaD77aMLZJadqxlDnwaAjO2EALw_wcB:G:s&amp;s_kwcid=AL!10441!3!793469273490!e!!g!!university%20of%20canberra!20950742786!163508319248&amp;gad_source=1&amp;gad_campaignid=20950742786&amp;gbraid=0AAAAADQmSqsslAu6v43ity0rQlfGofLnJ&amp;gclid=Cj0KCQjwnbrUBhDOARIsAKKhPpc52C3Utu5KibqLODSYHjrKrW0jG7_rQP9ghGaD77aMLZJadqxlDnwaAjO2EALw_wcB University of Canberra]. I plan to join the DFAT 2028 Graduate Program and become a diplomat representing Australia overseas. 

Currently, I'm watching [[wikipedia:The_Mentalist|The Mentalist]], and highly recommend it to anyone who enjoys a hearty, crime thriller!

== Current Book Chapter ==
[[Motivation and emotion/Book/2026/Need to love and be loved|Need to Love and be Loved]]

== Social Contributions ==
{{ic|Use a numbered list as shown in Tutorial 2}}

[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261?entry_id=809975 Discussed interest in learning about risky decision-making behaviours] (UCLearn Discussions Forum)

[[Talk:Motivation and emotion/Book/2023/Climate change emotion#c-U3280743-20260827072300-Feedback|Feedback on topic about climate change and motivation]] (Book chapter, 2023)

[https://en.wikiversity.org/w/index.php?diff=2827525 Grammatical error] (Book chapter, 2018)</text>
      <sha1>b1g1nikxx2c7jcrsxk6fq2ze9b5o9gp</sha1>
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  </page>
  <page>
    <title>User:U3260591</title>
    <ns>2</ns>
    <id>331570</id>
    <revision>
      <id>2829790</id>
      <parentid>2827206</parentid>
      <timestamp>2026-08-30T22:50:44Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Social contributions */</comment>
      <origin>2829790</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1269" sha1="n04csizdbhu5nxmczbab68yonmqhz5u" xml:space="preserve">== About me ==
Hi, I am a University of Canberra student currently studying Motivation and Emotion. I am interested in learning more about the psychological factors that influence motivation and behaviour, particularly in relation to exercise and physical activity. This is because one day, I wish to make my own business involving both motivation and physical activity.

== Book chapter I'm working on ==
I am currently developing a book chapter about [[Motivation and emotion/Book/2026/Exercise gamification motivation|Exercise gamification motivation]], which explores how gamification can affect exercise motivation and behaviour.

== Social contributions ==
{{ic|Use a numbered list}}

[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FMotivations_for_using_sex_work_services&amp;diff=2827161&amp;oldid=2827036 Corrected &quot;has&quot; to &quot;have&quot; to improve readability]

[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457809 Posted about my book chapter in UCLearn Discussion Forum]

[https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2026/Motivations_for_using_sex_work_services&amp;action=edit&amp;section=3 Added a comment to 'Motivations for using sex work services' to guide 'Stress and Trauma' section]</text>
      <sha1>n04csizdbhu5nxmczbab68yonmqhz5u</sha1>
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  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Oxytocin as a neuromodulator</title>
    <ns>0</ns>
    <id>331573</id>
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      <id>2829862</id>
      <parentid>2827713</parentid>
      <timestamp>2026-08-31T09:56:51Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Copyediting</comment>
      <origin>2829862</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17009" sha1="3wz2r4c877q2u8draxelroap4552jh6" xml:space="preserve">{{title|Oxytocin as a neuromodulator:&lt;br&gt;What are the motivational and emotional effects of oxytocin as a neuromodulator?}}
__TOC__

=== Overview ===
{{RoundBoxTop|theme=4}}

[[File:Oxytocin.svg|alt=Calvero|thumb|'''Figure 1'''. Chemical structure of oxytocin. ]]

'''Case study: Sarah and the power of connection'''

Sarah recently moved away from home to begin university in a new city. During her first few weeks, she feels nervous and isolated and often avoids social situations. Eventually, she forms a friendship with another student who makes her feel comfortable and supported. Over time, Sarah notices that she feels calmer around her new friend and begins looking forward to spending time with them. She also becomes more motivated to attend social events, meet other students, and form new relationships.

Sarah's experience raises an interesting question: why can positive social connections influence both how we feel and how motivated we are to interact with others? 

{{RoundBoxBottom}}

Oxytocin is a chemical messenger that acts as both a hormone in the body and a neuromodulator in the brain. Although it is commonly associated with bonding and affection, its role is more complex. Oxytocin can influence brain systems involved in social behaviour, emotional processing, stress, reward, and motivation.

Understanding these effects is important because social relationships can have a powerful influence on human behaviour and emotional experiences. Psychological and neuroscience research can help explain how oxytocin contributes to behaviours such as seeking social connection, forming attachments, responding to social rewards, and experiencing emotions such as trust, anxiety, and fear. Examining oxytocin as a neuromodulator can therefore provide insight into the biological processes that contribute to motivation and emotion. 

{{RoundBoxTop|theme=2}}
;Focus questions 

{{ic|Use bullet points as shown in Tutorial 2}}

1. How does oxytocin function as a neuromodulator in the brain?

2. How does oxytocin influence social motivation and reward?

3. How does oxytocin influence emotional processes such as trust, anxiety, and fear?

4. What factors influence the motivational and emotional effects of oxytocin?

{{RoundBoxBottom}}

==Headings ==
* [[#Overview|Overview]]
* Oxytocin as a neuromodulator'''
** What is oxytocin?
** How does oxytocin act in the brain?
** Brain regions and neural systems involved
* Oxytocin and motivation
** Social motivation and affiliation
** Reward and approach behaviour
** Bonding and attachment
*Oxytocin and emotion
** Stress and anxiety
** Fear and emotional processing
** Trust, empathy, and social emotions  '''4. Context and individual differences'''
** Social context
** Individual differences
** Limitations and mixed findings
* [[#Conclusion|Conclusion]] 
* [[#See also|See also]]
* [[#References|References]]
* [[#External links|External links]]

==Key points ==
'''Overview'''
* Introduce oxytocin as a neuropeptide that can function as both a hormone and a neuromodulator in the brain (Jurek &amp; Neumann, 2018).
* Introduce how oxytocin is associated with motivational and emotional processes, particularly social behaviour, bonding, reward, stress, and emotional processing (Gordon et al., 2011).
* Use the opening scenario to demonstrate how social connection may influence both emotional experiences and motivation to seek social interaction.
* Introduce the focus questions that will guide the chapter.

'''Oxytocin as a neuromodulator'''

* Oxytocin is produced primarily by neurons in the hypothalamus and can act both peripherally and within the central nervous system (Jurek &amp; Neumann, 2018).
* As a neuromodulator, oxytocin can alter neural activity rather than producing one simple behavioural response.
* Oxytocin receptors are found in neural systems involved in social behaviour, emotion, motivation, and reward (Donaldson &amp; Young, 2008).

'''What is oxytocin?'''

* Oxytocin is a nine-amino-acid neuropeptide with both hormonal and neuromodulatory functions (Jurek &amp; Neumann, 2018).
* Peripheral oxytocin is well known for its involvement in childbirth and lactation.
* Central oxytocin has broader functions involving social behaviour, bonding, stress, and emotional processes (Donaldson &amp; Young, 2008).

'''How does oxytocin act in the brain?'''

* Oxytocin binds to oxytocin receptors and modifies the activity of neurons and neural circuits.
* Its neuromodulatory effects can change how socially and emotionally relevant information is processed.
* Oxytocin interacts with other signalling systems, including dopamine pathways involved in motivation and reward (Love, 2014).

'''Brain regions and neural systems involved'''

* The hypothalamus contains important oxytocin-producing neurons and plays a central role in the oxytocin system (Jurek &amp; Neumann, 2018).
* Oxytocin can influence the amygdala, which is involved in processing emotionally and socially significant information.
* Oxytocin interacts with reward-related neural systems, including dopamine pathways associated with social motivation (Love, 2014).

----'''Oxytocin and motivation'''

* Oxytocin has been linked with social motivation and people's tendency to seek and maintain social relationships (Gordon et al., 2011).
* Oxytocin may influence how rewarding or important social stimuli are perceived.
* Its motivational effects depend on social context rather than universally increasing social approach (Shamay-Tsoory &amp; Abu-Akel, 2016).

'''Social motivation and affiliation'''

* Oxytocin may contribute to motivation to approach and interact with other people (Gordon et al., 2011).
* Oxytocin can increase the salience of social information, potentially influencing attention and subsequent behaviour (Shamay-Tsoory &amp; Abu-Akel, 2016).
* Social context and individual characteristics can determine whether increased social salience encourages approach or avoidance.

'''Reward and approach behaviour'''

* Oxytocin interacts with dopamine systems involved in motivation and reward (Love, 2014).
* Oxytocin may contribute to the rewarding nature of positive social interactions.
* Social reward may motivate individuals to repeat behaviours that promote social connection and affiliation.

'''Bonding and attachment'''

* Oxytocin has an established role in neural processes associated with social bonding and attachment (Donaldson &amp; Young, 2008).
* Interactions between oxytocin and reward pathways are important for the formation of social preferences and bonds (Young &amp; Wang, 2004).
* These processes may motivate people to seek proximity to and maintain relationships with significant social partners.

----'''Oxytocin and emotion'''

* Oxytocin can modulate neural systems involved in emotional processing, stress, and responses to social information.
* Research has examined its relationship with fear, anxiety, empathy, trust, and recognition of emotional information (Hurlemann et al., 2010).
* Oxytocin does not universally produce positive emotions; its emotional effects can depend on context and individual characteristics (Shamay-Tsoory &amp; Abu-Akel, 2016).

'''Stress and anxiety'''

* Oxytocin is involved in neural processes associated with regulation of stress and anxiety.
* Social support may interact with the oxytocin system and influence physiological and emotional responses to stressful experiences.
* The relationship between oxytocin and anxiety is complex and may depend on characteristics of both the person and the situation.

'''Fear and emotional processing'''

* Oxytocin can influence activity within the amygdala and related emotional-processing networks.
* It may alter how individuals respond to socially relevant emotional cues.
* These effects may contribute to differences in how threatening, safe, or emotionally important social situations are perceived.

'''Trust, empathy, and social emotions'''

* Oxytocin has been studied in relation to trust, empathy, and other social-emotional processes.
* Hurlemann et al. (2010) found that intranasal oxytocin enhanced emotional empathy and socially reinforced learning in healthy men.
* These findings suggest that oxytocin can influence the processing and significance of socially relevant emotional information.

----'''Context and individual differences'''

* Oxytocin's effects cannot be explained simply as increasing positive or prosocial behaviour.
* The social salience hypothesis proposes that oxytocin increases the importance of social cues, with behavioural outcomes depending on the situation (Shamay-Tsoory &amp; Abu-Akel, 2016).
* Individual characteristics can also contribute to differences in responses to oxytocin.

'''Social context'''

* The effects of oxytocin may differ between supportive, threatening, familiar, and unfamiliar social situations.
* Increasing the salience of social cues does not necessarily produce a positive behavioural response (Shamay-Tsoory &amp; Abu-Akel, 2016).
* Social context therefore needs to be considered when interpreting research about oxytocin and behaviour.

'''Individual differences'''

* Responses to oxytocin may vary according to biological and psychological characteristics.
* Previous social experiences and sensitivity to social information may influence responses.
* These differences may partly explain inconsistent findings across oxytocin studies.

'''Limitations and mixed findings'''

* Describing oxytocin as the '''&quot;love hormone&quot;''' oversimplifies its complex neuromodulatory effects.
* Findings from intranasal oxytocin studies should be interpreted carefully because administered oxytocin does not necessarily represent naturally occurring central oxytocin activity.
* Further research is needed to understand how dose, context, individual differences, and research methodology influence observed effects.

----'''Conclusion'''

* Oxytocin functions as a neuromodulator that can influence neural systems involved in both '''motivation and emotion'''.
* Its motivational effects include social affiliation, reward, approach behaviour, bonding, and attachment, while its emotional effects include modulation of stress, fear, anxiety, empathy, and other social emotions.
* Overall, oxytocin's effects are '''context-dependent and complex''', meaning it should not simply be understood as a chemical that produces love, trust, or positive social behaviour (Shamay-Tsoory &amp; Abu-Akel, 2016).
* Return to the opening scenario to demonstrate how oxytocin's neuromodulatory role may help explain connections between social experiences, emotional responses, and motivation.

==Figures==
[[File:Photographer taking a group photograph of smiling students in front of the Tokyo station, Marunouchi, Japan.jpg|thumb|'''Figure 2'''. Positive Social Interaction. {{ic|Explain in more detail; connect to text}}]]
Positive social interactions can influence feelings of connection and motivation to engage with others (see Figure 2).


* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure; this can be the figure in the scenario
* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* Images must be embedded from [[commons:|Wikimedia Commons]] which hosts free-to-use media such as photos, diagrams, graphs, video, and audio
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|For the final book chapter, I will include embedded Wiki links throughout the chapter to help readers understand important concepts and access further information. Key terms such as oxytocin, neuromodulation, dopamine, amygdala, motivation, emotion, social bonding, and reward will be linked to relevant Wikipedia and Wikiversity pages. These links will provide additional background information for readers who may be unfamiliar with the biological and psychological concepts discussed in the chapter.}}

;Embedded links 
For the final book chapter, I will include embedded Wiki links throughout the chapter to help readers understand important concepts and access further information. Key terms such as '''oxytocin, neuromodulation, dopamine, amygdala, motivation, emotion, social bonding, and reward''' will be linked to relevant Wikipedia and Wikiversity pages. These links will provide additional background information for readers who may be unfamiliar with the biological and psychological concepts discussed in the chapter.

{{anchor|Tables}}
;'''Table 1'''

* '''Motivational and emotional effects of oxytocin'''.
* Motivational effects could include social approach, affiliation, bonding, attachment, and social reward.
* Emotional effects could include stress, anxiety, fear, trust, empathy, and emotional processing.

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! Motivational effects !! Emotional effects 
|-
| Social Approach || Stress and anxiety 
|-
| Affiliation || Fear Processing 
|-
|Social Reward
|Trust
|-
|Bonding and attachment
|Empathy
|-
|Maintaining social relationships
|Emotional Processing 
|}

''Table 1'' 


==Conclusion==
** Oxytocin acts as a neuromodulator by influencing brain systems involved in '''social behaviour, reward, motivation, and emotional processing'''. Its interactions with neural systems, including dopamine and reward pathways, may help explain how social experiences influence motivated behaviour (Love, 2014).
** The motivational effects of oxytocin include its involvement in '''social approach, affiliation, bonding, attachment, and social reward'''. Oxytocin may increase the importance and rewarding value of social information, which can influence motivation to seek and maintain social relationships (Gordon et al., 2011).
** Oxytocin can also influence '''emotional processes''', including stress, anxiety, fear, empathy, trust, and responses to socially relevant emotional information. However, oxytocin should not simply be considered a “love hormone”, as it does not always produce positive emotional or social outcomes (Shamay-Tsoory &amp; Abu-Akel, 2016).
** Overall, the motivational and emotional effects of oxytocin are '''complex and context-dependent'''. Individual differences and the social environment can influence its effects, highlighting the importance of considering both biological and psychological factors when understanding oxytocin's role in human behaviour (Shamay-Tsoory &amp; Abu-Akel, 2016).
==See also==

* [[wikipedia:Emotion|Emotion]] (Wikipedia)
* [[wikipedia:Motivation|Motivation]] (Wikipedia)
* [[wikipedia:Neuromodulation|Neuromodulation]] (Wikipedia)
* [[wikipedia:Oxytocin|Oxytocin]] (Wikipedia)
* [[Motivation and emotion/Book/2024/Oxytocin and motivation|Oxytocin and motivation]] (Book chapter, 2024)

==References==
{{Hanging indent|1=

Donaldson, Z. R., &amp; Young, L. J. (2008). Oxytocin, vasopressin, and the neurogenetics of sociality. ''Science, 322''(5903), 900–904. https://doi.org/10.1126/science.1158668

Gordon, I., Martin, C., Feldman, R., &amp; Leckman, J. F. (2011). Oxytocin and social motivation. ''Developmental Cognitive Neuroscience, 1''(4), 471–493. https://doi.org/10.1016/j.dcn.2011.07.007

Hurlemann, R., Patin, A., Onur, O. A., Cohen, M. X., Baumgartner, T., Metzler, S., Dziobek, I., Gallinat, J., Wagner, M., Maier, W., &amp; Kendrick, K. M. (2010). Oxytocin enhances amygdala-dependent, socially reinforced learning and emotional empathy in humans. ''The Journal of Neuroscience, 30''(14), 4999–5007. https://doi.org/10.1523/JNEUROSCI.5538-09.2010

Jurek, B., &amp; Neumann, I. D. (2018). The oxytocin receptor: From intracellular signaling to behavior. ''Physiological Reviews, 98''(3), 1805–1908. https://doi.org/10.1152/physrev.00031.2017

Love, T. M. (2014). Oxytocin, motivation and the role of dopamine. ''Pharmacology, Biochemistry and Behavior, 119'', 49–60. https://doi.org/10.1016/j.pbb.2013.06.011

Shamay-Tsoory, S. G., &amp; Abu-Akel, A. (2016). The social salience hypothesis of oxytocin. ''Biological Psychiatry, 79''(3), 194–202. https://doi.org/10.1016/j.biopsych.2015.07.020

Young, L. J., &amp; Wang, Z. (2004). The neurobiology of pair bonding. ''Nature Neuroscience, 7''(10), 1048–1054. https://doi.org/10.1038/nn1327

}}

==External links==

* [https://www.ncbi.nlm.nih.gov/books/NBK507848/ Oxytocin] (National Library of Medicine)
* [https://www.yourhormones.info/hormones/oxytocin/ Oxytocin] (Society for Endocrinology)
* [https://www.health.harvard.edu/mind-and-mood/oxytocin-the-love-hormone Oxytocin: The love hormone?] (Harvard Health Publishing)

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__TOC__

=== Overview ===
{{RoundBoxTop|theme=4}}

[[File:Oxytocin.svg|alt=Calvero|thumb|'''Figure 1'''. Chemical structure of oxytocin. ]]

'''Case study: Sarah and the power of connection'''

Sarah recently moved away from home to begin university in a new city. During her first few weeks, she feels nervous and isolated and often avoids social situations. Eventually, she forms a friendship with another student who makes her feel comfortable and supported. Over time, Sarah notices that she feels calmer around her new friend and begins looking forward to spending time with them. She also becomes more motivated to attend social events, meet other students, and form new relationships.

Sarah's experience raises an interesting question: why can positive social connections influence both how we feel and how motivated we are to interact with others? 

{{RoundBoxBottom}}

Oxytocin is a chemical messenger that acts as both a hormone in the body and a neuromodulator in the brain. Although it is commonly associated with bonding and affection, its role is more complex. Oxytocin can influence brain systems involved in social behaviour, emotional processing, stress, reward, and motivation.

Understanding these effects is important because social relationships can have a powerful influence on human behaviour and emotional experiences. Psychological and neuroscience research can help explain how oxytocin contributes to behaviours such as seeking social connection, forming attachments, responding to social rewards, and experiencing emotions such as trust, anxiety, and fear. Examining oxytocin as a neuromodulator can therefore provide insight into the biological processes that contribute to motivation and emotion. 

{{RoundBoxTop|theme=2}}
;Focus questions 

{{ic|Use bullet points as shown in Tutorial 2}}

1. How does oxytocin function as a neuromodulator in the brain?

2. How does oxytocin influence social motivation and reward?

3. How does oxytocin influence emotional processes such as trust, anxiety, and fear?

4. What factors influence the motivational and emotional effects of oxytocin?

{{RoundBoxBottom}}

==Headings ==
* [[#Overview|Overview]]
* Oxytocin as a neuromodulator'''
** What is oxytocin?
** How does oxytocin act in the brain?
** Brain regions and neural systems involved
* Oxytocin and motivation
** Social motivation and affiliation
** Reward and approach behaviour
** Bonding and attachment
*Oxytocin and emotion
** Stress and anxiety
** Fear and emotional processing
** Trust, empathy, and social emotions  '''4. Context and individual differences'''
** Social context
** Individual differences
** Limitations and mixed findings
* [[#Conclusion|Conclusion]] 
* [[#See also|See also]]
* [[#References|References]]
* [[#External links|External links]]

==Key points ==
'''Overview'''
* Introduce oxytocin as a neuropeptide that can function as both a hormone and a neuromodulator in the brain (Jurek &amp; Neumann, 2018).
* Introduce how oxytocin is associated with motivational and emotional processes, particularly social behaviour, bonding, reward, stress, and emotional processing (Gordon et al., 2011).
* Use the opening scenario to demonstrate how social connection may influence both emotional experiences and motivation to seek social interaction.
* Introduce the focus questions that will guide the chapter.

'''Oxytocin as a neuromodulator'''

* Oxytocin is produced primarily by neurons in the hypothalamus and can act both peripherally and within the central nervous system (Jurek &amp; Neumann, 2018).
* As a neuromodulator, oxytocin can alter neural activity rather than producing one simple behavioural response.
* Oxytocin receptors are found in neural systems involved in social behaviour, emotion, motivation, and reward (Donaldson &amp; Young, 2008).

'''What is oxytocin?'''

* Oxytocin is a nine-amino-acid neuropeptide with both hormonal and neuromodulatory functions (Jurek &amp; Neumann, 2018).
* Peripheral oxytocin is well known for its involvement in childbirth and lactation.
* Central oxytocin has broader functions involving social behaviour, bonding, stress, and emotional processes (Donaldson &amp; Young, 2008).

'''How does oxytocin act in the brain?'''

* Oxytocin binds to oxytocin receptors and modifies the activity of neurons and neural circuits.
* Its neuromodulatory effects can change how socially and emotionally relevant information is processed.
* Oxytocin interacts with other signalling systems, including dopamine pathways involved in motivation and reward (Love, 2014).

'''Brain regions and neural systems involved'''

* The hypothalamus contains important oxytocin-producing neurons and plays a central role in the oxytocin system (Jurek &amp; Neumann, 2018).
* Oxytocin can influence the amygdala, which is involved in processing emotionally and socially significant information.
* Oxytocin interacts with reward-related neural systems, including dopamine pathways associated with social motivation (Love, 2014).

----'''Oxytocin and motivation'''

* Oxytocin has been linked with social motivation and people's tendency to seek and maintain social relationships (Gordon et al., 2011).
* Oxytocin may influence how rewarding or important social stimuli are perceived.
* Its motivational effects depend on social context rather than universally increasing social approach (Shamay-Tsoory &amp; Abu-Akel, 2016).

'''Social motivation and affiliation'''

* Oxytocin may contribute to motivation to approach and interact with other people (Gordon et al., 2011).
* Oxytocin can increase the salience of social information, potentially influencing attention and subsequent behaviour (Shamay-Tsoory &amp; Abu-Akel, 2016).
* Social context and individual characteristics can determine whether increased social salience encourages approach or avoidance.

'''Reward and approach behaviour'''

* Oxytocin interacts with dopamine systems involved in motivation and reward (Love, 2014).
* Oxytocin may contribute to the rewarding nature of positive social interactions.
* Social reward may motivate individuals to repeat behaviours that promote social connection and affiliation.

'''Bonding and attachment'''

* Oxytocin has an established role in neural processes associated with social bonding and attachment (Donaldson &amp; Young, 2008).
* Interactions between oxytocin and reward pathways are important for the formation of social preferences and bonds (Young &amp; Wang, 2004).
* These processes may motivate people to seek proximity to and maintain relationships with significant social partners.

----'''Oxytocin and emotion'''

* Oxytocin can modulate neural systems involved in emotional processing, stress, and responses to social information.
* Research has examined its relationship with fear, anxiety, empathy, trust, and recognition of emotional information (Hurlemann et al., 2010).
* Oxytocin does not universally produce positive emotions; its emotional effects can depend on context and individual characteristics (Shamay-Tsoory &amp; Abu-Akel, 2016).

'''Stress and anxiety'''

* Oxytocin is involved in neural processes associated with regulation of stress and anxiety.
* Social support may interact with the oxytocin system and influence physiological and emotional responses to stressful experiences.
* The relationship between oxytocin and anxiety is complex and may depend on characteristics of both the person and the situation.

'''Fear and emotional processing'''

* Oxytocin can influence activity within the amygdala and related emotional-processing networks.
* It may alter how individuals respond to socially relevant emotional cues.
* These effects may contribute to differences in how threatening, safe, or emotionally important social situations are perceived.

'''Trust, empathy, and social emotions'''

* Oxytocin has been studied in relation to trust, empathy, and other social-emotional processes.
* Hurlemann et al. (2010) found that intranasal oxytocin enhanced emotional empathy and socially reinforced learning in healthy men.
* These findings suggest that oxytocin can influence the processing and significance of socially relevant emotional information.

----'''Context and individual differences'''

* Oxytocin's effects cannot be explained simply as increasing positive or prosocial behaviour.
* The social salience hypothesis proposes that oxytocin increases the importance of social cues, with behavioural outcomes depending on the situation (Shamay-Tsoory &amp; Abu-Akel, 2016).
* Individual characteristics can also contribute to differences in responses to oxytocin.

'''Social context'''

* The effects of oxytocin may differ between supportive, threatening, familiar, and unfamiliar social situations.
* Increasing the salience of social cues does not necessarily produce a positive behavioural response (Shamay-Tsoory &amp; Abu-Akel, 2016).
* Social context therefore needs to be considered when interpreting research about oxytocin and behaviour.

'''Individual differences'''

* Responses to oxytocin may vary according to biological and psychological characteristics.
* Previous social experiences and sensitivity to social information may influence responses.
* These differences may partly explain inconsistent findings across oxytocin studies.

'''Limitations and mixed findings'''

* Describing oxytocin as the '''&quot;love hormone&quot;''' oversimplifies its complex neuromodulatory effects.
* Findings from intranasal oxytocin studies should be interpreted carefully because administered oxytocin does not necessarily represent naturally occurring central oxytocin activity.
* Further research is needed to understand how dose, context, individual differences, and research methodology influence observed effects.

----'''Conclusion'''

* Oxytocin functions as a neuromodulator that can influence neural systems involved in both '''motivation and emotion'''.
* Its motivational effects include social affiliation, reward, approach behaviour, bonding, and attachment, while its emotional effects include modulation of stress, fear, anxiety, empathy, and other social emotions.
* Overall, oxytocin's effects are '''context-dependent and complex''', meaning it should not simply be understood as a chemical that produces love, trust, or positive social behaviour (Shamay-Tsoory &amp; Abu-Akel, 2016).
* Return to the opening scenario to demonstrate how oxytocin's neuromodulatory role may help explain connections between social experiences, emotional responses, and motivation.

==Figures==
[[File:Photographer taking a group photograph of smiling students in front of the Tokyo station, Marunouchi, Japan.jpg|thumb|'''Figure 2'''. Positive Social Interaction. {{ic|Explain in more detail; connect to text}}]]
Positive social interactions can influence feelings of connection and motivation to engage with others (see Figure 2).


* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure; this can be the figure in the scenario
* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* Images must be embedded from [[commons:|Wikimedia Commons]] which hosts free-to-use media such as photos, diagrams, graphs, video, and audio
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|For the final book chapter, I will include embedded Wiki links throughout the chapter to help readers understand important concepts and access further information. Key terms such as oxytocin, neuromodulation, dopamine, amygdala, motivation, emotion, social bonding, and reward will be linked to relevant Wikipedia and Wikiversity pages. These links will provide additional background information for readers who may be unfamiliar with the biological and psychological concepts discussed in the chapter.}}

;Embedded links 
For the final book chapter, I will include embedded Wiki links throughout the chapter to help readers understand important concepts and access further information. Key terms such as '''oxytocin, neuromodulation, dopamine, amygdala, motivation, emotion, social bonding, and reward''' will be linked to relevant Wikipedia and Wikiversity pages. These links will provide additional background information for readers who may be unfamiliar with the biological and psychological concepts discussed in the chapter.

{{anchor|Tables}}
;'''Table 1'''

* '''Motivational and emotional effects of oxytocin'''.
* Motivational effects could include social approach, affiliation, bonding, attachment, and social reward.
* Emotional effects could include stress, anxiety, fear, trust, empathy, and emotional processing.

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! Motivational effects !! Emotional effects 
|-
| Social Approach || Stress and anxiety 
|-
| Affiliation || Fear Processing 
|-
|Social Reward
|Trust
|-
|Bonding and attachment
|Empathy
|-
|Maintaining social relationships
|Emotional Processing 
|}

''Table 1'' 


==Conclusion==
** Oxytocin acts as a neuromodulator by influencing brain systems involved in '''social behaviour, reward, motivation, and emotional processing'''. Its interactions with neural systems, including dopamine and reward pathways, may help explain how social experiences influence motivated behaviour (Love, 2014).
** The motivational effects of oxytocin include its involvement in '''social approach, affiliation, bonding, attachment, and social reward'''. Oxytocin may increase the importance and rewarding value of social information, which can influence motivation to seek and maintain social relationships (Gordon et al., 2011).
** Oxytocin can also influence '''emotional processes''', including stress, anxiety, fear, empathy, trust, and responses to socially relevant emotional information. However, oxytocin should not simply be considered a “love hormone”, as it does not always produce positive emotional or social outcomes (Shamay-Tsoory &amp; Abu-Akel, 2016).
** Overall, the motivational and emotional effects of oxytocin are '''complex and context-dependent'''. Individual differences and the social environment can influence its effects, highlighting the importance of considering both biological and psychological factors when understanding oxytocin's role in human behaviour (Shamay-Tsoory &amp; Abu-Akel, 2016).
==See also==

* [[wikipedia:Emotion|Emotion]] (Wikipedia)
* [[wikipedia:Motivation|Motivation]] (Wikipedia)
* [[wikipedia:Neuromodulation|Neuromodulation]] (Wikipedia)
* [[wikipedia:Oxytocin|Oxytocin]] (Wikipedia)
* [[Motivation and emotion/Book/2024/Oxytocin and motivation|Oxytocin and motivation]] (Book chapter, 2024)

==References==
{{Hanging indent|1=

Donaldson, Z. R., &amp; Young, L. J. (2008). Oxytocin, vasopressin, and the neurogenetics of sociality. ''Science, 322''(5903), 900–904. https://doi.org/10.1126/science.1158668

Gordon, I., Martin, C., Feldman, R., &amp; Leckman, J. F. (2011). Oxytocin and social motivation. ''Developmental Cognitive Neuroscience, 1''(4), 471–493. https://doi.org/10.1016/j.dcn.2011.07.007

Hurlemann, R., Patin, A., Onur, O. A., Cohen, M. X., Baumgartner, T., Metzler, S., Dziobek, I., Gallinat, J., Wagner, M., Maier, W., &amp; Kendrick, K. M. (2010). Oxytocin enhances amygdala-dependent, socially reinforced learning and emotional empathy in humans. ''The Journal of Neuroscience, 30''(14), 4999–5007. https://doi.org/10.1523/JNEUROSCI.5538-09.2010

Jurek, B., &amp; Neumann, I. D. (2018). The oxytocin receptor: From intracellular signaling to behavior. ''Physiological Reviews, 98''(3), 1805–1908. https://doi.org/10.1152/physrev.00031.2017

Love, T. M. (2014). Oxytocin, motivation and the role of dopamine. ''Pharmacology, Biochemistry and Behavior, 119'', 49–60. https://doi.org/10.1016/j.pbb.2013.06.011

Shamay-Tsoory, S. G., &amp; Abu-Akel, A. (2016). The social salience hypothesis of oxytocin. ''Biological Psychiatry, 79''(3), 194–202. https://doi.org/10.1016/j.biopsych.2015.07.020

Young, L. J., &amp; Wang, Z. (2004). The neurobiology of pair bonding. ''Nature Neuroscience, 7''(10), 1048–1054. https://doi.org/10.1038/nn1327

}}

==External links==

* [https://www.ncbi.nlm.nih.gov/books/NBK507848/ Oxytocin] (National Library of Medicine)
* [https://www.yourhormones.info/hormones/oxytocin/ Oxytocin] (Society for Endocrinology)
* [https://www.health.harvard.edu/mind-and-mood/oxytocin-the-love-hormone Oxytocin: The love hormone?] (Harvard Health Publishing)

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__TOC__

=== Overview ===
{{RoundBoxTop|theme=4}}

[[File:Oxytocin.svg|alt=Calvero|thumb|'''Figure 1'''. Chemical structure of oxytocin. ]]

'''Case study: Sarah and the power of connection'''

Sarah recently moved away from home to begin university in a new city. During her first few weeks, she feels nervous and isolated and often avoids social situations. Eventually, she forms a friendship with another student who makes her feel comfortable and supported. Over time, Sarah notices that she feels calmer around her new friend and begins looking forward to spending time with them. She also becomes more motivated to attend social events, meet other students, and form new relationships.

Sarah's experience raises an interesting question: why can positive social connections influence both how we feel and how motivated we are to interact with others? 

{{RoundBoxBottom}}

Oxytocin is a chemical messenger that acts as both a hormone in the body and a neuromodulator in the brain. Although it is commonly associated with bonding and affection, its role is more complex. Oxytocin can influence brain systems involved in social behaviour, emotional processing, stress, reward, and motivation.

Understanding these effects is important because social relationships can have a powerful influence on human behaviour and emotional experiences. Psychological and neuroscience research can help explain how oxytocin contributes to behaviours such as seeking social connection, forming attachments, responding to social rewards, and experiencing emotions such as trust, anxiety, and fear. Examining oxytocin as a neuromodulator can therefore provide insight into the biological processes that contribute to motivation and emotion. 

{{RoundBoxTop|theme=2}}
;Focus questions 

{{ic|Use bullet points as shown in Tutorial 2}}

1. How does oxytocin function as a neuromodulator in the brain?

2. How does oxytocin influence social motivation and reward?

3. How does oxytocin influence emotional processes such as trust, anxiety, and fear?

4. What factors influence the motivational and emotional effects of oxytocin?

{{RoundBoxBottom}}

==Headings ==
* [[#Overview|Overview]]
* Oxytocin as a neuromodulator'''
** What is oxytocin?
** How does oxytocin act in the brain?
** Brain regions and neural systems involved
* Oxytocin and motivation
** Social motivation and affiliation
** Reward and approach behaviour
** Bonding and attachment
*Oxytocin and emotion
** Stress and anxiety
** Fear and emotional processing
** Trust, empathy, and social emotions  '''4. Context and individual differences'''
** Social context
** Individual differences
** Limitations and mixed findings
* [[#Conclusion|Conclusion]] 
* [[#See also|See also]]
* [[#References|References]]
* [[#External links|External links]]

==Key points ==
'''Overview'''
* Introduce oxytocin as a neuropeptide that can function as both a hormone and a neuromodulator in the brain (Jurek &amp; Neumann, 2018).
* Introduce how oxytocin is associated with motivational and emotional processes, particularly social behaviour, bonding, reward, stress, and emotional processing (Gordon et al., 2011).
* Use the opening scenario to demonstrate how social connection may influence both emotional experiences and motivation to seek social interaction.
* Introduce the focus questions that will guide the chapter.

'''Oxytocin as a neuromodulator'''

* Oxytocin is produced primarily by neurons in the hypothalamus and can act both peripherally and within the central nervous system (Jurek &amp; Neumann, 2018).
* As a neuromodulator, oxytocin can alter neural activity rather than producing one simple behavioural response.
* Oxytocin receptors are found in neural systems involved in social behaviour, emotion, motivation, and reward (Donaldson &amp; Young, 2008).

'''What is oxytocin?'''

* Oxytocin is a nine-amino-acid neuropeptide with both hormonal and neuromodulatory functions (Jurek &amp; Neumann, 2018).
* Peripheral oxytocin is well known for its involvement in childbirth and lactation.
* Central oxytocin has broader functions involving social behaviour, bonding, stress, and emotional processes (Donaldson &amp; Young, 2008).

'''How does oxytocin act in the brain?'''

* Oxytocin binds to oxytocin receptors and modifies the activity of neurons and neural circuits.
* Its neuromodulatory effects can change how socially and emotionally relevant information is processed.
* Oxytocin interacts with other signalling systems, including dopamine pathways involved in motivation and reward (Love, 2014).

'''Brain regions and neural systems involved'''

* The hypothalamus contains important oxytocin-producing neurons and plays a central role in the oxytocin system (Jurek &amp; Neumann, 2018).
* Oxytocin can influence the amygdala, which is involved in processing emotionally and socially significant information.
* Oxytocin interacts with reward-related neural systems, including dopamine pathways associated with social motivation (Love, 2014).

----'''Oxytocin and motivation'''

* Oxytocin has been linked with social motivation and people's tendency to seek and maintain social relationships (Gordon et al., 2011).
* Oxytocin may influence how rewarding or important social stimuli are perceived.
* Its motivational effects depend on social context rather than universally increasing social approach (Shamay-Tsoory &amp; Abu-Akel, 2016).

'''Social motivation and affiliation'''

* Oxytocin may contribute to motivation to approach and interact with other people (Gordon et al., 2011).
* Oxytocin can increase the salience of social information, potentially influencing attention and subsequent behaviour (Shamay-Tsoory &amp; Abu-Akel, 2016).
* Social context and individual characteristics can determine whether increased social salience encourages approach or avoidance.

'''Reward and approach behaviour'''

* Oxytocin interacts with dopamine systems involved in motivation and reward (Love, 2014).
* Oxytocin may contribute to the rewarding nature of positive social interactions.
* Social reward may motivate individuals to repeat behaviours that promote social connection and affiliation.

'''Bonding and attachment'''

* Oxytocin has an established role in neural processes associated with social bonding and attachment (Donaldson &amp; Young, 2008).
* Interactions between oxytocin and reward pathways are important for the formation of social preferences and bonds (Young &amp; Wang, 2004).
* These processes may motivate people to seek proximity to and maintain relationships with significant social partners.

----'''Oxytocin and emotion'''

* Oxytocin can modulate neural systems involved in emotional processing, stress, and responses to social information.
* Research has examined its relationship with fear, anxiety, empathy, trust, and recognition of emotional information (Hurlemann et al., 2010).
* Oxytocin does not universally produce positive emotions; its emotional effects can depend on context and individual characteristics (Shamay-Tsoory &amp; Abu-Akel, 2016).

'''Stress and anxiety'''

* Oxytocin is involved in neural processes associated with regulation of stress and anxiety.
* Social support may interact with the oxytocin system and influence physiological and emotional responses to stressful experiences.
* The relationship between oxytocin and anxiety is complex and may depend on characteristics of both the person and the situation.

'''Fear and emotional processing'''

* Oxytocin can influence activity within the amygdala and related emotional-processing networks.
* It may alter how individuals respond to socially relevant emotional cues.
* These effects may contribute to differences in how threatening, safe, or emotionally important social situations are perceived.

'''Trust, empathy, and social emotions'''

* Oxytocin has been studied in relation to trust, empathy, and other social-emotional processes.
* Hurlemann et al. (2010) found that intranasal oxytocin enhanced emotional empathy and socially reinforced learning in healthy men.
* These findings suggest that oxytocin can influence the processing and significance of socially relevant emotional information.

----'''Context and individual differences'''

* Oxytocin's effects cannot be explained simply as increasing positive or prosocial behaviour.
* The social salience hypothesis proposes that oxytocin increases the importance of social cues, with behavioural outcomes depending on the situation (Shamay-Tsoory &amp; Abu-Akel, 2016).
* Individual characteristics can also contribute to differences in responses to oxytocin.

'''Social context'''

* The effects of oxytocin may differ between supportive, threatening, familiar, and unfamiliar social situations.
* Increasing the salience of social cues does not necessarily produce a positive behavioural response (Shamay-Tsoory &amp; Abu-Akel, 2016).
* Social context therefore needs to be considered when interpreting research about oxytocin and behaviour.

'''Individual differences'''

* Responses to oxytocin may vary according to biological and psychological characteristics.
* Previous social experiences and sensitivity to social information may influence responses.
* These differences may partly explain inconsistent findings across oxytocin studies.

'''Limitations and mixed findings'''

* Describing oxytocin as the '''&quot;love hormone&quot;''' oversimplifies its complex neuromodulatory effects.
* Findings from intranasal oxytocin studies should be interpreted carefully because administered oxytocin does not necessarily represent naturally occurring central oxytocin activity.
* Further research is needed to understand how dose, context, individual differences, and research methodology influence observed effects.

----'''Conclusion'''

* Oxytocin functions as a neuromodulator that can influence neural systems involved in both '''motivation and emotion'''.
* Its motivational effects include social affiliation, reward, approach behaviour, bonding, and attachment, while its emotional effects include modulation of stress, fear, anxiety, empathy, and other social emotions.
* Overall, oxytocin's effects are '''context-dependent and complex''', meaning it should not simply be understood as a chemical that produces love, trust, or positive social behaviour (Shamay-Tsoory &amp; Abu-Akel, 2016).
* Return to the opening scenario to demonstrate how oxytocin's neuromodulatory role may help explain connections between social experiences, emotional responses, and motivation.

==Figures==
[[File:Photographer taking a group photograph of smiling students in front of the Tokyo station, Marunouchi, Japan.jpg|thumb|'''Figure 2'''. Positive Social Interaction. {{ic|Explain in more detail; connect to text}}]]
Positive social interactions can influence feelings of connection and motivation to engage with others (see Figure 2).


* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure; this can be the figure in the scenario
* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* Images must be embedded from [[commons:|Wikimedia Commons]] which hosts free-to-use media such as photos, diagrams, graphs, video, and audio
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|For the final book chapter, I will include embedded Wiki links throughout the chapter to help readers understand important concepts and access further information. Key terms such as oxytocin, neuromodulation, dopamine, amygdala, motivation, emotion, social bonding, and reward will be linked to relevant Wikipedia and Wikiversity pages. These links will provide additional background information for readers who may be unfamiliar with the biological and psychological concepts discussed in the chapter.}}

;Embedded links 
For the final book chapter, I will include embedded Wiki links throughout the chapter to help readers understand important concepts and access further information. Key terms such as '''oxytocin, neuromodulation, dopamine, amygdala, motivation, emotion, social bonding, and reward''' will be linked to relevant Wikipedia and Wikiversity pages. These links will provide additional background information for readers who may be unfamiliar with the biological and psychological concepts discussed in the chapter.

{{anchor|Tables}}
;'''Table 1'''

* '''Motivational and emotional effects of oxytocin'''.
* Motivational effects could include social approach, affiliation, bonding, attachment, and social reward.
* Emotional effects could include stress, anxiety, fear, trust, empathy, and emotional processing.

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! Motivational effects !! Emotional effects 
|-
| Social Approach || Stress and anxiety 
|-
| Affiliation || Fear Processing 
|-
|Social Reward
|Trust
|-
|Bonding and attachment
|Empathy
|-
|Maintaining social relationships
|Emotional Processing 
|}

''Table 1'' 


==Conclusion==
** Oxytocin acts as a neuromodulator by influencing brain systems involved in '''social behaviour, reward, motivation, and emotional processing'''. Its interactions with neural systems, including dopamine and reward pathways, may help explain how social experiences influence motivated behaviour (Love, 2014).
** The motivational effects of oxytocin include its involvement in '''social approach, affiliation, bonding, attachment, and social reward'''. Oxytocin may increase the importance and rewarding value of social information, which can influence motivation to seek and maintain social relationships (Gordon et al., 2011).
** Oxytocin can also influence '''emotional processes''', including stress, anxiety, fear, empathy, trust, and responses to socially relevant emotional information. However, oxytocin should not simply be considered a “love hormone”, as it does not always produce positive emotional or social outcomes (Shamay-Tsoory &amp; Abu-Akel, 2016).
** Overall, the motivational and emotional effects of oxytocin are '''complex and context-dependent'''. Individual differences and the social environment can influence its effects, highlighting the importance of considering both biological and psychological factors when understanding oxytocin's role in human behaviour (Shamay-Tsoory &amp; Abu-Akel, 2016).
==See also==

* [[wikipedia:Emotion|Emotion]] (Wikipedia)
* [[wikipedia:Motivation|Motivation]] (Wikipedia)
* [[wikipedia:Neuromodulation|Neuromodulation]] (Wikipedia)
* [[wikipedia:Oxytocin|Oxytocin]] (Wikipedia)
* [[Motivation and emotion/Book/2024/Oxytocin and motivation|Oxytocin and motivation]] (Book chapter, 2024)

==References==
{{Hanging indent|1=

Donaldson, Z. R., &amp; Young, L. J. (2008). Oxytocin, vasopressin, and the neurogenetics of sociality. ''Science, 322''(5903), 900–904. https://doi.org/10.1126/science.1158668

Gordon, I., Martin, C., Feldman, R., &amp; Leckman, J. F. (2011). Oxytocin and social motivation. ''Developmental Cognitive Neuroscience, 1''(4), 471–493. https://doi.org/10.1016/j.dcn.2011.07.007

Hurlemann, R., Patin, A., Onur, O. A., Cohen, M. X., Baumgartner, T., Metzler, S., Dziobek, I., Gallinat, J., Wagner, M., Maier, W., &amp; Kendrick, K. M. (2010). Oxytocin enhances amygdala-dependent, socially reinforced learning and emotional empathy in humans. ''The Journal of Neuroscience, 30''(14), 4999–5007. https://doi.org/10.1523/JNEUROSCI.5538-09.2010

Jurek, B., &amp; Neumann, I. D. (2018). The oxytocin receptor: From intracellular signaling to behavior. ''Physiological Reviews, 98''(3), 1805–1908. https://doi.org/10.1152/physrev.00031.2017

Love, T. M. (2014). Oxytocin, motivation and the role of dopamine. ''Pharmacology, Biochemistry and Behavior, 119'', 49–60. https://doi.org/10.1016/j.pbb.2013.06.011

Shamay-Tsoory, S. G., &amp; Abu-Akel, A. (2016). The social salience hypothesis of oxytocin. ''Biological Psychiatry, 79''(3), 194–202. https://doi.org/10.1016/j.biopsych.2015.07.020

Young, L. J., &amp; Wang, Z. (2004). The neurobiology of pair bonding. ''Nature Neuroscience, 7''(10), 1048–1054. https://doi.org/10.1038/nn1327

}}

==External links==

* [https://www.ncbi.nlm.nih.gov/books/NBK507848/ Oxytocin] (National Library of Medicine)
* [https://www.yourhormones.info/hormones/oxytocin/ Oxytocin] (Society for Endocrinology)
* [https://www.health.harvard.edu/mind-and-mood/oxytocin-the-love-hormone Oxytocin: The love hormone?] (Harvard Health Publishing)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Hormones/Oxytocin]]
[[Category:Motivation and emotion/Book/Neuromodulators/Cortisol]]</text>
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  <page>
    <title>Talk:Motivation and emotion/Book/2026/Retirement motivation</title>
    <ns>1</ns>
    <id>331593</id>
    <revision>
      <id>2829829</id>
      <parentid>2828114</parentid>
      <timestamp>2026-08-31T04:14:19Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Topic development feedback</comment>
      <origin>2829829</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6748" sha1="iwndsbspr6m8evt3xrtav9n6koizbnm" xml:space="preserve">== Great start! ==

I really like the way you've structured your chapter so far. The consideration of intrinsic and extrinsic motivation is interesting and very relevant. I am curious about how this has changed over generations? Is there more desire to have life balance out of work these days? Looking forward to seeing this progress.  [[User:Tammysaurus|Tammysaurus]] ([[User talk:Tammysaurus|discuss]] • [[Special:Contributions/Tammysaurus|contribs]]) 10:28, 27 August 2026 (UTC)

== Heading casing ==

I noticed in your chapter on Self-determination theory that you've capitalised each word for this title.

The recommended heading casing for [[Wikiversity:Introduction|Wikiversity]] uses [[wikipedia:Letter_case#Sentence_case|sentence casing]]. As an example:

&lt;big&gt;Self-determination theory&lt;/big&gt; rather than &lt;big&gt;Self-Determination Theory.&lt;/big&gt;

Here's an example of that sentence casing: [[Self-determination theory]]

Sorry if that's annoying and I hope it helps! [[User:U3292769|U3292769]] ([[User talk:U3292769|discuss]] • [[Special:Contributions/U3292769|contribs]]) 12:16, 27 August 2026 (UTC)

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement
# Consider making a clearer connection to motivation
&lt;!-- Alignment with focus questions --&gt;
# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
# &quot;factors&quot; is vague; maybe try &quot;motivational factrs&quot;
# This is a motivation topic; insofaras emotional topics are considered, they should be in relation to motivation
&lt;!-- GenAI ---&gt;
# Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
|3=
&lt;!-- Overview--&gt;
# Scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- GenAI ---&gt;
# Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Scenario --&gt;
# The scenario could be more tailored to focus on motivational aspects of retirement
&lt;!-- Description --&gt;
# Strive for a user-friendly overview of the motivational problem and key theory/research
# Use plain English; most of the citations can be moved into subsequent sections
&lt;!-- Focus questions --&gt;
# Good alignment between focus questions and heading structure, but consider closer alignment
# Clarify &quot;factors&quot;
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Key points are well developed for each section
# The intial section lacks clear theoretical grounding; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title
&lt;!-- Theory and research --&gt;
# Select the best theories about this topic
# Select the best research about this topic
# Self-determination theory should be in sentence casing (APA style)
&lt;!-- GenAI ---&gt;
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is underway
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Relevant figure(s) presented, captioned, and cited
&lt;!-- Creation --&gt;
# Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
&lt;!-- Scenarios/examples/case studies --&gt;
# Promising use of scenarios/examples/case studies
&lt;!-- Quiz --&gt;
# Promising use of quiz question(s)
&lt;!-- Tables --&gt;
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Very good
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
# Only reference sources that you have consulted
&lt;!-- APA style --&gt;
# Check and correct capitalisation and italicisation
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Good
### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]]
&lt;!-- External links --&gt;
# External links
## Australians represent &lt; 0.5% of the world's population
## Link to the most relevant international/global external resources about this topic
|9=
&lt;!-- User page --&gt;
# Good
# Very brief description about self – consider expanding
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# Provide an internal link to the book chapter (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
|10=
&lt;!-- Social contribution --&gt;
# Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:14, 31 August 2026 (UTC)</text>
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  <page>
    <title>Talk:Motivation and emotion/Book/2026/Need to love and be loved</title>
    <ns>1</ns>
    <id>331631</id>
    <revision>
      <id>2829842</id>
      <parentid>2828962</parentid>
      <timestamp>2026-08-31T07:23:28Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Topic development feedback</comment>
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      <format>text/x-wiki</format>
      <text bytes="5413" sha1="9lvtaphd7vpqsvb0umfbvc2wv3x4e1p" xml:space="preserve">== Maslow's Hierarchy of Needs ==

Hi, I think adding a visual of Maslow's Hierarchy of needs would be very beneficial here as it can be difficult to describe in writing. Just a simple pyramid style would be great! Something like this: https://images.ctfassets.net/pdf29us7flmy/5DmjqTuW3FuMY4Hbtm8Xj0/6e4607bb76080f18663f19ecd6adaafc/maslow-s-hierarchy-of-needs-new.png?w=1440&amp;q=100&amp;fm=avif  [[User:E3297976|E3297976]] ([[User talk:E3297976|discuss]] • [[Special:Contributions/E3297976|contribs]]) 22:30, 27 August 2026 (UTC)

== The distinct nature of love vs belonging ==

Hello, excellent work I like the way you have set up your chapter. I do posit a more robust distinction and explanation or comparison between the sense of belonging one feels adjacent to the expression of love as described. Perhaps shedding light on how love is its own emotional capacity different to belonging would be interesting to divulge into. I hope my suggestion makes sense. 

Regards. ￼ [[User:Mort006|Mort006]] ([[User talk:Mort006|discuss]] • [[Special:Contributions/Mort006|contribs]]) 06:27, 28 August 2026 (UTC)

== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:U3280743|Hi [[User:U3280743|U3280743]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:23, 31 August 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed)
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
# Remove numbers from headings
&lt;-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
&lt;!-- Alignment with focus questions --&gt;
# Insufficient alignment between sub-title, focus questions, and top-level headings
|3=
&lt;!-- Overview--&gt;
&lt;!-- Scenario --&gt;
# Make the relevance of the scenario to the topic more clear
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
&lt;!-- Focus questions --&gt;
# Excellent focus questions
# Develop closer alignment between the focus questions and top-level headings
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Promising development
# Highlight the most relevant theories and synthesise the best research on the topic
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is about right
&lt;!-- Theory and research --&gt;
# Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. 
&lt;!-- Conclusion --&gt;
# Conclusion is underway
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Relevant figure(s) are presented and captioned
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Promising use of scenarios/examples/case studies
# Make the scenario directly relevant to a key point about the best theory and research
&lt;!-- Quiz --&gt;
# Consider including quiz question(s) about the take-home messages
&lt;!-- Tables --&gt;
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Very good
&lt;!-- Systematic reviews --&gt;
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## some titles are incomplete
## page numbers should be separated by an en-dash (–) rather than a hyphen (-)
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
&lt;!-- External links --&gt;
# External links
## Very good
## Use [[w:Letter case#Sentence casing|sentence casing]]
|9=
&lt;!-- User page --&gt;
# Basic but effective
&lt;!-- Description about self --&gt;
# Very brief description about self – consider expanding
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
|10=
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:23, 31 August 2026 (UTC)</text>
      <sha1>9lvtaphd7vpqsvb0umfbvc2wv3x4e1p</sha1>
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  </page>
  <page>
    <title>Introductory Ancient Greek Language/Lesson 12</title>
    <ns>0</ns>
    <id>331642</id>
    <revision>
      <id>2829747</id>
      <parentid>2829650</parentid>
      <timestamp>2026-08-30T18:16:21Z</timestamp>
      <contributor>
        <username>It-is-Truly-Meet</username>
        <id>3089598</id>
      </contributor>
      <minor />
      <comment>/* Principal Parts */</comment>
      <origin>2829747</origin>
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      <format>text/x-wiki</format>
      <text bytes="5586" sha1="p53a3tb2hlp34yh2yglp9aif99wsk87" xml:space="preserve">== The Aorist ==
The aorist is used to denote a one-time completed action, like French's {{w|passé simple}}. While both the imperfect and aorist refer to past events, they differ in aspect: The '''aorist''' always conveys a '''discreet''' action, while the '''imperfect''' tense always conveys past activity that was '''more than a single action''' in some way.
* '''Imperfect''': καὶ '''ἤρχοντο''' πρὸς αὐτὸν καὶ ἔλεγον· Χαῖρε, ὁ βασιλεὺς τῶν Ἰουδαίων· — and [they] '''were coming''' up to him again, saying, “Hail, king of the Jews!”
* '''Aorist''': καὶ '''ἐξῆλθεν''' πάλιν ἔξω ὁ Πιλᾶτος καὶ λέγει αὐτοῖς — &quot;Once more Pilate '''came''' out and said to the [Jews gathered there]&quot;

The aorist and imperfect are '''secondary''' tenses, so an '''augment''' precedes the stem in the indicative mood, and they both use '''secondary endings'''. However, the two tenses use '''different''' stems: The imperfect generally uses the present tense stem for any given verb; the aorist almost always uses the verb stem.
* '''Present''': λαμβαν
* '''Aorist''': λαβ

===1st and 2nd Aorist===
The 1st aorist adds -σα- to the verb stem; the 2nd aorist omits it. When the secondary endings for –μι verbs were added to the first aorist marker -σα-, the 1st aorist endings evolved:
{| class=&quot;wikitable&quot; style=&quot;text-align:center&quot;
! Singular
! Dual
! Plural
|-
| 1. -σα- + -ν = -σα
| colspan=2|-σα- + -μεν = -σαμεν
|-
| 2. -σα- + -ς = -σας
| -σα- + -τον = -σατον
| -σα- + -τε = -σατε
|-
| 3. -σα- + — = -σε
| -σα- + -την = -σατην
| -σα- + -σαν = -σαν
|}
Take the aorist active indicative of '''δείκνυμι''' as an example:
{| class=&quot;wikitable&quot;
| ἔδειξα
| colspan=2|ἐδείξαμεν
|-
| ἔδειξας
| ἐδείξατον
| ἐδείξατε
|-
| ἔδειξε(ν)
| ἐδειξάτην
| ἔδειξαν
|}
Remember: κ + σ = ξ

====Liquid and Nasal 1st Aorists====
For 1st aorists of '''liquid''' (λ, ρ) or '''nasal''' stems (μ, ν), the σ marker drops (the same fact with liquid futures), and its loss often leads to '''compensatory lengthening''' (e.g., ε often lengthens to ει).
*  ἀγγέλλω ('''stem''': ἀγγελ-): ἤγγελ&lt;big&gt;'''σ'''&lt;/big&gt;α → ἤγγειλα

===The 2nd Aorist===
The two types of 2nd aorists are '''thematic''', the most common, and athematic, which use the same secondary endings.

Secondary indicative stem: λαβ → ἐλαβ
{| class=&quot;wikitable&quot;
| ἔλαβον
| colspan=2|ἐλάβομεν
|-
| ἔλαβες
| ἐλάβετον
| ἐλάβετε
|-
| ἔλαβε(ν)
| ἐλαβέτην
| ἔλαβον
|}

====Athematic 2nd Aorist====
There are not many athematic 2nd aorists; only two are commonly encountered:
* βαίνω, βήσομαι, ἔβην (verb stem: βη-): walk, come, go
* γιγνώσκω, γνώσομαι, ἔγνων (verb stem: γνω-): know, learn, think

== Infinitives and Aspect ==
The aorist, like the present and future tenses, occurs in the infinitive mood. The augment to secondary tenses indicates '''actual historical action'''; thus only the '''indicative''' mood uses it. The aorist infinitive, a verbal noun, '''never''' has the augment.

===1st Aorist Infinitive===
'''Formation''': verb stem + σαι.
The persistent '''accent''' falls on the '''penult''', the second-to-last syllable.
* δεῖ.ξαι
* πισ.τεῦ.σαι
* γε.νη.θῆ.ναι

===2nd Aorist Infinitive===
'''Thematic formation''': verb stem + ειν. The '''present''' indicative active infinitive accents on the '''penult''' (e.g., λαμβάνειν); the 2nd aorist active indicative infinitive has a '''circumflex''' on the '''ultima''' (e.g., λαβεῖν).

'''Αthematic formation''': verb stem + ναι. The athematic 2nd aorist uses the same infinitive ending as the athematic present tense. Like the present tense, the accent falls on the '''penult'''.
* γνῶ.ναι
* βῆ.ναι

===Infinitive and Aspect===
If an infinitive is used as an '''articular''' or a '''complementary infinitive''', the present and aorist tenses of the infinitive are used to express '''aspect''', not time. In other words, the present infinitive expresses ongoing activity, while the aorist infinitive expresses a simple momentary action. This distinction is often unexpressed in English.

* '''Present infinitive''': παίζειν βούλομαι. — I want to be playing.
* '''Aorist infinitive''': παῖσαι βούλομαι. — I want to play.

== Principal Parts ==
Verbs are alphabetized by their 1st person, singular, '''present''' active indicative form, with a -μι or -ω ending. This is the '''first principal part'''.

The '''second principal part''' is the 1st person singular, '''future''' active indicative. This form is necessary because adding -σ- to the verb stem can result in some unexpected forms.

The '''third principal part''' is the 1st person singular, '''aorist''' active indicative. The third principal part of a verb shows whether it forms a first or second aorist, and if the latter, whether it is thematic or athematic.
* βουλεύω, βουλεύσω, ἐβούλευσα (1st Aorist)
* φέρω, οἴσω, ἤνεγκα (1st Aorist)
* λείπω, λείψω, ἔλιπον (2nd Aorist Thematic)
* τρέχω, δραμοῦμαι, ἔδραμον (2nd Aorist Thematic)
* βαίνω, βήσομαι, ἔβην (2nd Aorist Athematic)

[[Category:Ancient Greek Language]]</text>
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  </page>
  <page>
    <title>Universal Bibliography/Cinema</title>
    <ns>0</ns>
    <id>331713</id>
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      <timestamp>2026-08-30T16:57:06Z</timestamp>
      <contributor>
        <username>James500</username>
        <id>297601</id>
      </contributor>
      <comment>/* */ Add</comment>
      <origin>2829740</origin>
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      <text bytes="21643" sha1="q3rdi6jye2n2gbfrm8elga3eo84m162" xml:space="preserve">{{Bibliography}}
See also [[w:Bibliography of film by genre]], [[w:List of books on films]], [[w:Category:Books about film]] and [[s:Category:Works about film]]

This part of the [[Universal Bibliography]] is a bibliography of cinema.

Bibliography
*George Rehrauer. The Macmillan Film Bibliography. [https://books.google.com/books?id=DmcoAAAAMAAJ]
*Malte Hagener and Michael Töteberg. Film – An International Bibliography. Verlag JB Metzler. 2002. [https://books.google.co.uk/books?id=4r28DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*James Monaco and Susan Schenker. Books about Film: A Bibliographical Checklist. New York Zoetrope. 1976. [https://books.google.com/books?id=-xzgAAAAMAAJ]
*Jack C Ellis, Charles Derry and Sharon Kern. The Film Book Bibliography, 1940-1975. Scarecrow Press. 1979. [https://books.google.com/books?id=BnWuAAAAIAAJ]
*Robert A Armour. Film: A Reference Guide. 1980. [https://books.google.co.uk/books?id=8X5ZAAAAMAAJ]
*Frank Manchel. Film Study: An Analytical Bibliography. 1990. [https://books.google.co.uk/books?id=BebEAji_wH4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=ID4E3Lm8TsgC&amp;pg=PA953#v=onepage&amp;q&amp;f=false vol 2].

General
*Magill's Survey of Cinema

Series
*Screen Series. A Zwemmer Limited, London. A S Barnes &amp; Co. New York. (eg Arne Svensson. Japan. 1971. [https://books.google.co.uk/books?id=W4RZAAAAMAAJ])
*World Cinema. Flicks Books. [https://books.google.co.uk/books?id=Im9ZAAAAMAAJ vol 1]. [https://books.google.co.uk/books?id=7TxNAQAAIAAJ vol 2]. [https://books.google.co.uk/books?id=pS9AAQAAIAAJ vol 4]. [https://books.google.co.uk/books?id=AvjoDrhajOcC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 5]. [https://books.google.co.uk/books?id=yAInAQAAIAAJ]

Periodicals

See also [[w:List of film periodicals]]
*East-West Film Journal [https://books.google.co.uk/books?id=3T8bAQAAIAAJ] (began December 1986)

Annuals
*Magill's Cinema Annual. [https://books.google.co.uk/books?id=LXAsctC3wasC 1983] (1982 films). [https://books.google.co.uk/books?id=V5XSO0pJmUUC 1986] (1985 films).
*Picturegoer Film Annual. [https://books.google.co.uk/books?id=yhM6AQAAIAAJ for 1950-1951]. 
*The Motion Picture Annual. [https://books.google.co.uk/books?id=UhUIAQAAMAAJ 1989].
*Boy's Cinema Annual. Amalgamated Press. [https://books.google.co.uk/books?id=QVfych4dj54C 1939].
*International Film Annual [https://books.google.co.uk/books?id=D4I3AAAAIAAJ No 3]
*The British Film Annual. Winchester Publications. [https://books.google.co.uk/books?id=QSMPAQAAMAAJ 1949].
*The Western Film Annual [https://books.google.co.uk/books?id=_kwoXXrpJU4C]

Yearbooks
*Film Daily Year Book of Motion Pictures. [https://books.google.co.uk/books?id=qJMHAQAAIAAJ 1969].

Years
*Edgar Anstey, &quot;The Cinema&quot; (1944) 172 The Spectator 10 (No 6028: 7 January 1944). Includes &quot;Review of the Year&quot;.

Reading
*James Monaco. How to Read a Film: The Art, Technology, Language, History, and Theory of Film and Media. 1981. [https://books.google.co.uk/books?id=inRZAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

World and national
*Kevin Rockett and John Hill. National Cinemas and World Cinema. 2006. [https://books.google.co.uk/books?id=pudkAAAAMAAJ]

World
*Elkan Allan (comp). A Guide to World Cinema. Whittet Books. 1985. [https://books.google.co.uk/books?id=nb8dAAAAMAAJ]
*Rob Stone, Paul Cooke, Stephanie Dennison and Alex Marlow-Mann (eds). The Routledge Companion to World Cinema. [https://books.google.co.uk/books?id=DnE3DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Geoffrey Nowell-Smith. The Oxford History of World Cinema. 1996. Paperback. 1997. [https://books.google.co.uk/books?id=MZwVDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*David Robinson. World Cinema: A Short History. Eyre Methuen. 1973. [https://books.google.co.uk/books?id=EH9ZAAAAMAAJ]
*William Luhr (ed). World Cinema Since 1945. Ungar. New York. 1987. [https://books.google.co.uk/books?id=6gAqAAAAYAAJ]
*James Chapman. Cinemas of the World: Film and Society from 1895 to the Present. 2003. [https://books.google.co.uk/books?id=SMYo4Abel2EC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Linda Badley (ed). Traditions in World Cinema. 2006. [https://books.google.co.uk/books?id=tpYkDQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kate Gamm. Teaching World Cinema. British Film Institute. 2004. [https://books.google.co.uk/books?id=PmJZAAAAMAAJ]
*Shekhar Deshpande and Meta Mazaj. World Cinema: A Critical Introduction. 2018. [https://books.google.co.uk/books?id=qFFHDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*John Hill and Pamela Church Gibson. World Cinema: Critical Approaches. 2000. [https://books.google.co.uk/books?id=SUKcAQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lúcia Nagib, Chris Perriam and Rajinder Dudrah (eds). Theorizing World Cinema. 2012. [https://books.google.co.uk/books?id=cgNQEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Asian
*Tom Vick. Asian Cinema: A Field Guide. HarperCollins. [https://books.google.co.uk/books?id=YpIaAQAAIAAJ]
*Aaron Han Joon Magnan-Park, Gina Marchetti and See Kam Tan (eds). The Palgrave Handbook of Asian Cinema. 2018. [https://books.google.co.uk/books?id=MQ92DwAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Zhen Zhang, Sangjoon Lee, Debashree Mukherjee and Intan Paramaditha (eds). The Routledge Companion to Asian Cinemas. 2024. [https://books.google.co.uk/books?id=bwYDEQAAQBAJ&amp;pg=PA1995#v=onepage&amp;q&amp;f=false]
*Dimitris Eleftheriotis and Gary Needham. Asian Cinemas: A Reader and Guide. 2006. [https://books.google.co.uk/books?id=tPqgEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Olivia Khoo. Asian Cinema: A Regional View. 2021. [https://books.google.co.uk/books?id=vvCgEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Stephen Teo. The Asian Cinema Experience: Styles, spaces, theory. Routledge. 2013. [https://books.google.co.uk/books?id=2sHZz-XUR2oC#v=onepage&amp;q&amp;f=false]
*John A Lent. The Asian Film Industry. University of Texas Press. 1990. [https://books.google.co.uk/books?id=JkkqAAAAYAAJ]
*Yau Shuk-ting and Kinnia (eds). East Asian Cinema and Cultural Heritage: From China, Hong Kong, Taiwan to Japan and South Korea. 2011. [https://books.google.co.uk/books?id=UeLHAAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Routledge Handbook of South Asian Cinemas. 2026. [https://books.google.co.uk/books?id=QEyWEQAAQBAJ&amp;pg=PA6#v=onepage&amp;q&amp;f=false]
*Jose F Lacaba (ed). The Films of ASEAN. 2000. [https://books.google.co.uk/books?id=DI6dq64VNDkC]

Periodicals, Asian cinema
*Asian Cinema [https://books.google.co.uk/books?id=W4saAQAAIAAJ]
*[[w:Cinemaya|Cinemaya]] [https://books.google.co.uk/books?id=3pMHAQAAIAAJ]
*South Asian Cinema [https://books.google.co.uk/books?id=0eVkAAAAMAAJ] (began February 2001)

Annuals, Asian cinema
*Asian Film Directory &amp; Who's Who. [https://books.google.co.uk/books?id=0wY6AQAAIAAJ 1952]

India

Hindi

See [[w:Bibliography of Hindi cinema]]

Japanese and Korean
*Justin Bowyer. The Cinema of Japan &amp; Korea. 2004. [https://books.google.co.uk/books?id=FtqUAxsisk0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kate E Taylor-Jones. Rising Sun, Divided Land: Japanese and South Korean Filmmakers. 2013. [https://books.google.co.uk/books?id=SfaEAAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals, Japanese and Korean
*Journal of Japanese and Korean Cinema [https://www.tandfonline.com/journals/rjkc20]

==Japanese==
*Arne Svensson. Japan. (Screen Series). A Zwemmer Limited, London. A S Barnes &amp; Co. New York. 1971. [https://books.google.co.uk/books?id=W4RZAAAAMAAJ]
*Daisuke Miyao (ed). The Oxford Handbook of Japanese Cinema. 2014. [https://books.google.co.uk/books?id=XXD1AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Joanne Bernardi, Shota T Ogawa (eds). Routledge Handbook of Japanese Cinema. 2021. [https://books.google.co.uk/books?id=xkX1DwAAQBAJ&amp;pg=PA1931#v=onepage&amp;q&amp;f=false]
*Thomas Weisser and Yuko Mihara Weisser. Japanese Cinema: The Essential Handbook. [https://books.google.com/books?id=IrZcSAAACAAJ]
*David Desser. A Companion to Japanese Cinema. 2022. [https://books.google.co.uk/books?id=YWl6EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Hideaki Fujiki and Alastair Phillips (eds). The Japanese Cinema Book. Bloomsbury, for British Film Institute. 2020. [https://books.google.co.uk/books?id=Zu3cDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Stuart Galbraith and Paul Duncan. Japanese Cinema. Taschen. 2009. ISBN 9783822831564. [https://books.google.co.uk/books?id=rDkFKAAACAAJ]. Catalogue: [https://catalogue.nla.gov.au/catalog/4701384]
*&quot;Japanese Cinema&quot;. Pam Cook (ed). The Cinema Book. British Film Institute. 1985. 3rd Ed: 2007. [https://books.google.co.uk/books?id=TTr8DwAAQBAJ&amp;pg=PA238#v=onepage&amp;q&amp;f=false p 238].
*Martha P Nochimson. &quot;Japan: Screening Feudalism and Modernism&quot;. World on Film: An Introduction. Chapter 5. [https://books.google.co.uk/books?id=DXK5gkQwAs8C&amp;pg=PA195#v=onepage&amp;q&amp;f=false p 195].
*Donald Richie. Japanese Cinema: An Introduction. Oxford University Press.  Oxford and New York. 1990. [https://books.google.co.uk/books?id=k4lZAAAAMAAJ]. Review: Cinemaya [https://books.google.co.uk/books?id=7ZEHAQAAIAAJ]
*Marie Seton, &quot;Japanese Cinema&quot; (1958) [https://books.google.co.uk/books?id=Y5WQS46o0a4C 1] The Living Cinema 189 (No 4: Spring 1958)
*Donald Richie. Japanese Cinema: Film Style and National Character. 1971. [https://books.google.com/books?id=0YlZAAAAMAAJ]
*Joan Mellen. The Waves at Genji's Door: Japan Through Its Cinema. Pantheon Books. 1976. [https://books.google.co.uk/books?id=alC0AAAAIAAJ]
*Joseph L Anderson and Donald Richie. The Japanese Film: Art and Industry. Tuttle. 1959. Evergreen. 1960. Expanded Edition. Princeton University Press. 1982. [https://books.google.co.uk/books?id=C2z3otM-y5kC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Stuart Galbraith IV. The Japanese Filmography: A Complete Reference to 209 Filmmakers and the Over 1250 Films Released in the United States, 1900 through 1994. 1996. [https://books.google.co.uk/books?id=GLxlEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Berra (ed). Directory of World Cinema: Japan. 2010. [https://books.google.co.uk/books?id=tQHsEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
**Directory of World Cinema: Japan 3. 2015. [https://books.google.co.uk/books?id=9QLsEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Abé Markus Nornes and Aaron Gerow. Research Guide to Japanese Film Studies. 2009. [https://books.google.co.uk/books?id=5UONCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Keiko I McDonald. Reading a Japanese Film: Cinema in Context. 2006. [https://books.google.co.uk/books?id=ICqSfjUqIpMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*David Bordwell, &quot;Our Dream Cinema: Western Historiography and the Japanese Film&quot; (1979) Film Reader, [https://books.google.co.uk/books?id=bp-RAAAAIAAJ No 4], p 45.

*Mitsuyo Wada-Marciano. Japanese Cinema in the Digital Age. 2012. [https://books.google.co.uk/books?id=3F4EEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Arthur Nolletti and David Desser. Reframing Japanese Cinema: Authorship, Genre, History. Indiana University Press. 1992. [https://books.google.co.uk/books?id=6Q1jAAAAMAAJ]
*Laura Lee. Japanese Cinema Between Frames. 2017. [https://books.google.co.uk/books?id=c8I9DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Noel Burch. To the Distant Observer: Form and Meaning in the Japanese Cinema. 1979. [https://books.google.co.uk/books?id=lLtyz275-GYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Dennis Washburn and Carole Cavanaugh (eds). Word and Image in Japanese Cinema. 2001. [https://books.google.co.uk/books?id=CwZ6SPpmJwQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Daisuke Miyao. The Aesthetics of Shadow: Lighting and Japanese Cinema. 2013. [https://books.google.co.uk/books?id=80K2AgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Justin Vicari. Japanese Film and the Floating Mind: Cinematic Contemplations of Being. 2016. [https://books.google.co.uk/books?id=CYuwDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Keiko I McDonald. Cinema East: A Critical Study of Major Japanese Films. Fairleigh Dickinson University Press. Associated University Presses. 1983. [https://books.google.co.uk/books?id=ywxjAAAAMAAJ]
*Alastair Phillips and Julian Stringer. Japanese Cinema: Texts and Contexts. 2007. [https://books.google.co.uk/books?id=53xRa5dwUOUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darrell William Davis. Picturing Japaneseness: Monumental Style, National Identity, Japanese Film. Columbia University Press. 1996. ISBN 0-231-10231-3. Review: [https://www.cambridge.org/core/journals/journal-of-asian-studies/article/abs/picturing-japaneseness-monumental-style-national-identity-japanese-film-by-darrell-william-davis-new-york-columbia-university-press-1996-viii-304-pp-1750/7CC73AD51C9A06CA40DF3463CC9D3E7D]

Bibliography
*Nornes and Gerow. &quot;Annotated Bibliography for Bibliographic Studies&quot;. Research Guide to Japanese Film Studies. Chapter 4. [https://books.google.co.uk/books?id=5UONCwAAQBAJ&amp;pg=PA72#v=onepage&amp;q&amp;f=false p 72].

Reference
*[https://guides.library.yale.edu/c.php?g=295932&amp;p=1973050 Japanese Reference Materials for Studying Japanese Cinema at Yale University]. Yale Library.

Periodicals

See also [[w:ja:Category:日本の映画雑誌]]. [映画雑誌 = film magazine] [映画機関誌 = film journal]
*Honchi Haruhiko (Japanese: 本地陽彦). Nihon Eiga Zasshi Taitoru Soran (Japanese: 日本映画雑誌タイトル総覧). [[w:ja:ワイズ出版|Waizu Shuppan]]. Tokyo. 2003. [https://books.google.co.uk/books?id=kskvAQAAIAAJ]. Commentary: Research Guide to Japanese Film Studies, [https://books.google.co.uk/books?id=5UONCwAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false p 83].
*[https://www.nfaj.go.jp/exhibition/filmmagazines/ 映画雑誌の秘かな愉しみ The Discreet Charm of Film Magazines]. National Film Archive of Japan (NFAJ).
*[https://guides.library.yale.edu/c.php?g=295932&amp;p=1973061 Japanese Reference Materials for Studying Japanese Cinema at Yale University: Film Periodicals]. Yale Library.
*[[w:en:Kinema Junpo|Kinema Junpo]] (Japanese: キネマ旬報) (The Movie Times) [https://books.google.co.uk/books?id=UFKtYBwc5ioC]
*[[w:en:UniJapan Film Quarterly|UniJapan Film Quarterly]] [https://books.google.co.uk/books?id=QIY3AAAAIAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/AA12490478]

Annuals and year books
*Japanese Films. UniJapan Film. [https://books.google.co.uk/books?id=4LAaAQAAIAAJ 1960]
**Japanese Film [https://books.google.co.uk/books?id=iJUHAQAAIAAJ 1983] [https://books.google.co.uk/books?id=fpYHAQAAIAAJ]
*Cinema Year Book of Japan. [https://books.google.co.uk/books?id=r7W9DZnBbBwC 1938].

History
*Jasper Sharp. Historical Dictionary of Japanese Cinema. 2011. [https://books.google.co.uk/books?id=YQR4EQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Isolde Standish. A New History of Japanese Cinema: A Century of Narrative Film. 2006. [https://books.google.co.uk/books?id=GZxUv4g_icQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Donald Ritchie. A Hundred Years of Japanese Film: A Concise History, with a Selective Guide to DVDs and Videos. 2001. 2005. [https://books.google.co.uk/books?id=s7-_Gon5-a0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yomota Inuhiko. What Is Japanese Cinema? A History. 2014. 2019. [https://books.google.co.uk/books?id=8Il-DwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Gerald Mast and Bruce F Kawin. &quot;Japan&quot; in &quot;Cinemas East&quot;. A Short History of the Movies. Allyn and Bacon. 7th Ed: 2000: [https://books.google.co.uk/books?id=AC0IAQAAMAAJ]. pp 403 to 417.
*Aaron Gerow. &quot;From Misemono to Zigomar: A Discursive History of Early Japanese Cinema&quot;. Bean, Kapse and Horak (eds). Silent Cinema and the Politics of Space. 2014. Chapter 6. [https://books.google.co.uk/books?id=7poiAwAAQBAJ&amp;pg=PA157#v=onepage&amp;q&amp;f=false p 157].

20th century
*Beverley Bare Buehrer. Japanese Films: A Filmography and Commentary, 1921-1989. McFarland &amp; Company. 1990. [https://books.google.co.uk/books?id=N6RZAAAAMAAJ]
*Aristides Gazetas. &quot;Post-War Japanese Cinema: 1950-1990&quot;. An Introduction to World Cinema. 2nd Ed. 2008. Chapter 13. [https://books.google.co.uk/books?id=CPuZ-2UtVRwC&amp;pg=PA169#v=onepage&amp;q&amp;f=false p 169].
*Japanese Experimental Film, 1960-1980. American Federation of Arts. [https://books.google.co.uk/books?id=XGUbAQAAIAAJ]
*Yuriko Furuhata. Cinema of Actuality: Japanese Avant-Garde Filmmaking in the Season of Image Politics. 2013. [https://books.google.co.uk/books?id=sDi2AgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [covers 1960s and 1970s].
*Outlaw Masters of Japanese Film. 2005. [https://books.google.co.uk/books?id=MbaKDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [covers 1950s to 1970s]
*Japanese Cinema: From Kurosawa To Tora-San&quot; (1986) Information Bulletin, February 1986, p 15 [https://books.google.co.uk/books?id=Sw5PAQAAIAAJ]
*Catherine Russell. Classical Japanese Cinema Revisited. 2011. [https://books.google.co.uk/books?id=uv9GAQAAQBAJ&amp;pg=PR4#v=onepage&amp;q&amp;f=false] [covers roughly 1930 to 1960]
*Kyoko Hirano. Mr. Smith Goes to Tokyo: Japanese Cinema Under the American Occupation, 1945-1952. 1992. [https://books.google.co.uk/books?id=6OsKAQAAMAAJ]
*Peter B High. The Imperial Screen: Japanese Film Culture in the Fifteen Years' War, 1931-1945. 2003. [https://books.google.co.uk/books?id=6XiA9DOuvjAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Naoki Yamamoto. Dialectics without Synthesis: Japanese Film Theory and Realism in a Global Frame. 2020. [https://books.google.co.uk/books?id=lEnrDwAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false] [Covers 1910s to 1950s].
*Sean O'Reilly (ed). The Advent of Sound in Japanese Cinema: A Handbook. 2025. [https://books.google.co.uk/books?id=vAKbEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [covers the 1930s]
*Mitsuyo Wada-Marciano. Nippon Modern: Japanese Cinema of the 1920s and 1930s. 2008. [https://books.google.co.uk/books?id=dIc0RBYMs9kC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Iwamoto Kenji, &quot;Japanese Cinema Until 1930: A Consideration of its Formal Aspects&quot;. Iris: A Journal of Theory on Image and Sound. No 16, p 9 [https://books.google.co.uk/books?id=zXFhCc7ENAEC]
*David Bordwell, &quot;Visual Style in Japanese Cinema, 1925-1945&quot; (1995) [https://books.google.co.uk/books?id=YqIqAQAAIAAJ 7] Film History 5 to 31 (No 1: Spring 1995)
*Sean D O'Reilly. Re-Viewing the Past: The Uses of History in the Cinema of Imperial Japan. 2018. Paperback Ed: 2020. [https://books.google.co.uk/books?id=fqFiDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

New, contemporary
*Mark Schilling. Contemporary Japanese Film. 1999. [https://books.google.co.uk/books?id=LNQ4EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kawamoto Saburo. New Trends in Japanese Cinema. (Orientation Seminars on Japan, No 21). The Japan Foundation, Office for the Japanese Studies Center. [https://books.google.co.uk/books?id=R5IqAQAAIAAJ]
*Tadao Sato. Currents in Japanese Cinema. Kodansha International. 1982. [https://books.google.com/books?id=p-9kAAAAMAAJ]

Genres
*Abe Mark Nornes. Japanese Documentary Film: The Meiji Era Through Hiroshima. 2003. [https://books.google.co.uk/books?id=2JFrwF2LMw4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*S A Thornton. The Japanese Period Film: A Critical Analysis. 2008. [https://books.google.co.uk/books?id=yhPo95wPz8QC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Alain Silver. The Samurai Film. 1977: [https://books.google.co.uk/books?id=R4BZAAAAMAAJ]. The Overlook Press. 1983: [https://books.google.co.uk/books?id=AH1ZAAAAMAAJ]
*David Desser. The Samurai Films of Akira Kurosawa. UMI Research Press. [https://books.google.com/books?id=AaJZAAAAMAAJ]
*Silver Screen Samurai: The Best of Japan's Samurai Movie Posters. 2004. [https://books.google.co.uk/books?id=tdsuoae7X68C&amp;pg=PA2#v=onepage&amp;q&amp;f=false]
*Thomas Weisser and Yuko Mihara Weisser. Japanese Cinema Encyclopedia: The Horror, Fantasy, and SciFi films. Vital Books. 1997. [https://books.google.com/books?id=HGSybwAACAAJ]
*Jay McRoy. Japanese Horror Cinema. University of Hawaii Press. Edinburgh University Press. 2005. [https://books.google.co.uk/books?id=XvOgEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] Review: Timothy Iles (2007) [https://books.google.co.uk/books?id=rt9KAQAAIAAJ 33] The Journal of Japanese Studies 264
*Colette Balmain. Introduction to Japanese Horror Film. 2008. [https://books.google.co.uk/books?id=POWqBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Michael Crandol. Ghost in the Well: The Hidden History of Horror Films in Japan. 2021. [https://books.google.co.uk/books?id=mDgfEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Peter C Pugsley. Japanese High School Films: Iconography, Nostalgia and Discipline. 2022. [https://books.google.co.uk/books?id=jPOgEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Peter C Pugsley. Beyond the High School Film: Reaching Adulthood in Everyday Japanese Cinema. 2026. [https://books.google.co.uk/books?id=7boEEgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cult
*Patrick Macias. Tokyoscope: The Japanese Cult Film Companion. 2001. [https://books.google.co.uk/books?id=g-0HAQAAMAAJ]

Directors
*Alexander Jacoby. A Critical Handbook of Japanese Film Directors: From the Silent Era to the Present Day. 2008. [https://books.google.co.uk/books?id=RhWNAgAAQBAJ&amp;pg=PA1870#v=onepage&amp;q&amp;f=false]
*Audie Bock. Japanese Film Directors. 1978. Paperback Ed. Kodansha International. 1985. [https://books.google.co.uk/books?id=A6EqAAAAYAAJ]
*Mitsuhiro Yoshimoto. Kurosawa: Film Studies and Japanese Cinema. 2000. [https://books.google.co.uk/books?id=QizaCOjKs-IC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Stars
*Hideaki Fujiki. Making Personas: Transnational Film Stardom in Modern Japan. 2013. [https://books.google.co.uk/books?id=A_gFEAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

Studios
*Stuart Galbraith IV. The Toho Studios Story: A History and Complete Filmography. 2008. [https://books.google.co.uk/books?id=f7o8pq6G_dYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Interviews
*Joan Mellen. Voices from the Japanese Cinema. Liveright. New York. 1975. ISBN 0871406047.

V-Cinema
*Tom Mes. Japanese Film and the Challenge of Video. 2023. [https://books.google.co.uk/books?id=70i7EAAAQBAJ&amp;pg=PA1921#v=onepage&amp;q&amp;f=false]

Film criticism
*Scott Nygren. Time Frames: Japanese Cinema And the Unfolding of History. [https://books.google.co.uk/books?id=PvlgQgAACAAJ]

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      <text bytes="21813" sha1="mkhqmcdrescrbyfw20v8ssu2n0ibl2w" xml:space="preserve">{{Bibliography}}
See also [[w:Bibliography of film by genre]], [[w:List of books on films]], [[w:Category:Books about film]] and [[s:Category:Works about film]]

This part of the [[Universal Bibliography]] is a bibliography of cinema.

Bibliography
*George Rehrauer. The Macmillan Film Bibliography. [https://books.google.com/books?id=DmcoAAAAMAAJ]
*Malte Hagener and Michael Töteberg. Film – An International Bibliography. Verlag JB Metzler. 2002. [https://books.google.co.uk/books?id=4r28DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*James Monaco and Susan Schenker. Books about Film: A Bibliographical Checklist. New York Zoetrope. 1976. [https://books.google.com/books?id=-xzgAAAAMAAJ]
*Jack C Ellis, Charles Derry and Sharon Kern. The Film Book Bibliography, 1940-1975. Scarecrow Press. 1979. [https://books.google.com/books?id=BnWuAAAAIAAJ]
*Robert A Armour. Film: A Reference Guide. 1980. [https://books.google.co.uk/books?id=8X5ZAAAAMAAJ]
*Frank Manchel. Film Study: An Analytical Bibliography. 1990. [https://books.google.co.uk/books?id=BebEAji_wH4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=ID4E3Lm8TsgC&amp;pg=PA953#v=onepage&amp;q&amp;f=false vol 2].

General
*Magill's Survey of Cinema

Series
*Screen Series. A Zwemmer Limited, London. A S Barnes &amp; Co. New York. (eg Arne Svensson. Japan. 1971. [https://books.google.co.uk/books?id=W4RZAAAAMAAJ])
*World Cinema. Flicks Books. [https://books.google.co.uk/books?id=Im9ZAAAAMAAJ vol 1]. [https://books.google.co.uk/books?id=7TxNAQAAIAAJ vol 2]. [https://books.google.co.uk/books?id=pS9AAQAAIAAJ vol 4]. [https://books.google.co.uk/books?id=AvjoDrhajOcC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 5]. [https://books.google.co.uk/books?id=yAInAQAAIAAJ]

Periodicals

See also [[w:List of film periodicals]]
*East-West Film Journal [https://books.google.co.uk/books?id=3T8bAQAAIAAJ] (began December 1986)

Annuals
*Magill's Cinema Annual. [https://books.google.co.uk/books?id=LXAsctC3wasC 1983] (1982 films). [https://books.google.co.uk/books?id=V5XSO0pJmUUC 1986] (1985 films).
*Picturegoer Film Annual. [https://books.google.co.uk/books?id=yhM6AQAAIAAJ for 1950-1951]. 
*The Motion Picture Annual. [https://books.google.co.uk/books?id=UhUIAQAAMAAJ 1989].
*Boy's Cinema Annual. Amalgamated Press. [https://books.google.co.uk/books?id=QVfych4dj54C 1939].
*International Film Annual [https://books.google.co.uk/books?id=D4I3AAAAIAAJ No 3]
*The British Film Annual. Winchester Publications. [https://books.google.co.uk/books?id=QSMPAQAAMAAJ 1949].
*The Western Film Annual [https://books.google.co.uk/books?id=_kwoXXrpJU4C]

Yearbooks
*Film Daily Year Book of Motion Pictures. [https://books.google.co.uk/books?id=qJMHAQAAIAAJ 1969].

Years
*Edgar Anstey, &quot;The Cinema&quot; (1944) 172 The Spectator 10 (No 6028: 7 January 1944). Includes &quot;Review of the Year&quot;.

Reading
*James Monaco. How to Read a Film: The Art, Technology, Language, History, and Theory of Film and Media. 1981. [https://books.google.co.uk/books?id=inRZAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

World and national
*Kevin Rockett and John Hill. National Cinemas and World Cinema. 2006. [https://books.google.co.uk/books?id=pudkAAAAMAAJ]

World
*Elkan Allan (comp). A Guide to World Cinema. Whittet Books. 1985. [https://books.google.co.uk/books?id=nb8dAAAAMAAJ]
*Rob Stone, Paul Cooke, Stephanie Dennison and Alex Marlow-Mann (eds). The Routledge Companion to World Cinema. [https://books.google.co.uk/books?id=DnE3DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Geoffrey Nowell-Smith. The Oxford History of World Cinema. 1996. Paperback. 1997. [https://books.google.co.uk/books?id=MZwVDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*David Robinson. World Cinema: A Short History. Eyre Methuen. 1973. [https://books.google.co.uk/books?id=EH9ZAAAAMAAJ]
*William Luhr (ed). World Cinema Since 1945. Ungar. New York. 1987. [https://books.google.co.uk/books?id=6gAqAAAAYAAJ]
*James Chapman. Cinemas of the World: Film and Society from 1895 to the Present. 2003. [https://books.google.co.uk/books?id=SMYo4Abel2EC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Linda Badley (ed). Traditions in World Cinema. 2006. [https://books.google.co.uk/books?id=tpYkDQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kate Gamm. Teaching World Cinema. British Film Institute. 2004. [https://books.google.co.uk/books?id=PmJZAAAAMAAJ]
*Shekhar Deshpande and Meta Mazaj. World Cinema: A Critical Introduction. 2018. [https://books.google.co.uk/books?id=qFFHDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*John Hill and Pamela Church Gibson. World Cinema: Critical Approaches. 2000. [https://books.google.co.uk/books?id=SUKcAQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lúcia Nagib, Chris Perriam and Rajinder Dudrah (eds). Theorizing World Cinema. 2012. [https://books.google.co.uk/books?id=cgNQEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Asian
*Tom Vick. Asian Cinema: A Field Guide. HarperCollins. [https://books.google.co.uk/books?id=YpIaAQAAIAAJ]
*Aaron Han Joon Magnan-Park, Gina Marchetti and See Kam Tan (eds). The Palgrave Handbook of Asian Cinema. 2018. [https://books.google.co.uk/books?id=MQ92DwAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Zhen Zhang, Sangjoon Lee, Debashree Mukherjee and Intan Paramaditha (eds). The Routledge Companion to Asian Cinemas. 2024. [https://books.google.co.uk/books?id=bwYDEQAAQBAJ&amp;pg=PA1995#v=onepage&amp;q&amp;f=false]
*Dimitris Eleftheriotis and Gary Needham. Asian Cinemas: A Reader and Guide. 2006. [https://books.google.co.uk/books?id=tPqgEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Olivia Khoo. Asian Cinema: A Regional View. 2021. [https://books.google.co.uk/books?id=vvCgEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Stephen Teo. The Asian Cinema Experience: Styles, spaces, theory. Routledge. 2013. [https://books.google.co.uk/books?id=2sHZz-XUR2oC#v=onepage&amp;q&amp;f=false]
*John A Lent. The Asian Film Industry. University of Texas Press. 1990. [https://books.google.co.uk/books?id=JkkqAAAAYAAJ]
*Yau Shuk-ting and Kinnia (eds). East Asian Cinema and Cultural Heritage: From China, Hong Kong, Taiwan to Japan and South Korea. 2011. [https://books.google.co.uk/books?id=UeLHAAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Routledge Handbook of South Asian Cinemas. 2026. [https://books.google.co.uk/books?id=QEyWEQAAQBAJ&amp;pg=PA6#v=onepage&amp;q&amp;f=false]
*Jose F Lacaba (ed). The Films of ASEAN. 2000. [https://books.google.co.uk/books?id=DI6dq64VNDkC]

Periodicals, Asian cinema
*Asian Cinema [https://books.google.co.uk/books?id=W4saAQAAIAAJ]
*[[w:Cinemaya|Cinemaya]] [https://books.google.co.uk/books?id=3pMHAQAAIAAJ]
*South Asian Cinema [https://books.google.co.uk/books?id=0eVkAAAAMAAJ] (began February 2001)

Annuals, Asian cinema
*Asian Film Directory &amp; Who's Who. [https://books.google.co.uk/books?id=0wY6AQAAIAAJ 1952]

India

Hindi

See [[w:Bibliography of Hindi cinema]]

Japanese and Korean
*Justin Bowyer. The Cinema of Japan &amp; Korea. 2004. [https://books.google.co.uk/books?id=FtqUAxsisk0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kate E Taylor-Jones. Rising Sun, Divided Land: Japanese and South Korean Filmmakers. 2013. [https://books.google.co.uk/books?id=SfaEAAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals, Japanese and Korean
*Journal of Japanese and Korean Cinema [https://www.tandfonline.com/journals/rjkc20]

==Japanese==
*Arne Svensson. Japan. (Screen Series). A Zwemmer Limited, London. A S Barnes &amp; Co. New York. 1971. [https://books.google.co.uk/books?id=W4RZAAAAMAAJ]
*Daisuke Miyao (ed). The Oxford Handbook of Japanese Cinema. 2014. [https://books.google.co.uk/books?id=XXD1AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Joanne Bernardi, Shota T Ogawa (eds). Routledge Handbook of Japanese Cinema. 2021. [https://books.google.co.uk/books?id=xkX1DwAAQBAJ&amp;pg=PA1931#v=onepage&amp;q&amp;f=false]
*Thomas Weisser and Yuko Mihara Weisser. Japanese Cinema: The Essential Handbook. [https://books.google.com/books?id=IrZcSAAACAAJ]
*David Desser. A Companion to Japanese Cinema. 2022. [https://books.google.co.uk/books?id=YWl6EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Hideaki Fujiki and Alastair Phillips (eds). The Japanese Cinema Book. Bloomsbury, for British Film Institute. 2020. [https://books.google.co.uk/books?id=Zu3cDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Stuart Galbraith and Paul Duncan. Japanese Cinema. Taschen. 2009. ISBN 9783822831564. [https://books.google.co.uk/books?id=rDkFKAAACAAJ]. Catalogue: [https://catalogue.nla.gov.au/catalog/4701384]
*&quot;Japanese Cinema&quot;. Pam Cook (ed). The Cinema Book. British Film Institute. 1985. 3rd Ed: 2007. [https://books.google.co.uk/books?id=TTr8DwAAQBAJ&amp;pg=PA238#v=onepage&amp;q&amp;f=false p 238].
*Martha P Nochimson. &quot;Japan: Screening Feudalism and Modernism&quot;. World on Film: An Introduction. Chapter 5. [https://books.google.co.uk/books?id=DXK5gkQwAs8C&amp;pg=PA195#v=onepage&amp;q&amp;f=false p 195].
*Donald Richie. Japanese Cinema: An Introduction. Oxford University Press.  Oxford and New York. 1990. [https://books.google.co.uk/books?id=k4lZAAAAMAAJ]. Review: Cinemaya [https://books.google.co.uk/books?id=7ZEHAQAAIAAJ]
*Marie Seton, &quot;Japanese Cinema&quot; (1958) [https://books.google.co.uk/books?id=Y5WQS46o0a4C 1] The Living Cinema 189 (No 4: Spring 1958)
*Donald Richie. Japanese Cinema: Film Style and National Character. 1971. [https://books.google.com/books?id=0YlZAAAAMAAJ]
*Joan Mellen. The Waves at Genji's Door: Japan Through Its Cinema. Pantheon Books. 1976. [https://books.google.co.uk/books?id=alC0AAAAIAAJ]
*Joseph L Anderson and Donald Richie. The Japanese Film: Art and Industry. Tuttle. 1959. Evergreen. 1960. Expanded Edition. Princeton University Press. 1982. [https://books.google.co.uk/books?id=C2z3otM-y5kC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Stuart Galbraith IV. The Japanese Filmography: A Complete Reference to 209 Filmmakers and the Over 1250 Films Released in the United States, 1900 through 1994. 1996. [https://books.google.co.uk/books?id=GLxlEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Berra (ed). Directory of World Cinema: Japan. 2010. [https://books.google.co.uk/books?id=tQHsEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
**Directory of World Cinema: Japan 3. 2015. [https://books.google.co.uk/books?id=9QLsEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Abé Markus Nornes and Aaron Gerow. Research Guide to Japanese Film Studies. 2009. [https://books.google.co.uk/books?id=5UONCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Keiko I McDonald. Reading a Japanese Film: Cinema in Context. 2006. [https://books.google.co.uk/books?id=ICqSfjUqIpMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*David Bordwell, &quot;Our Dream Cinema: Western Historiography and the Japanese Film&quot; (1979) Film Reader, [https://books.google.co.uk/books?id=bp-RAAAAIAAJ No 4], p 45.

*Mitsuyo Wada-Marciano. Japanese Cinema in the Digital Age. 2012. [https://books.google.co.uk/books?id=3F4EEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Arthur Nolletti and David Desser. Reframing Japanese Cinema: Authorship, Genre, History. Indiana University Press. 1992. [https://books.google.co.uk/books?id=6Q1jAAAAMAAJ]
*Laura Lee. Japanese Cinema Between Frames. 2017. [https://books.google.co.uk/books?id=c8I9DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Noel Burch. To the Distant Observer: Form and Meaning in the Japanese Cinema. 1979. [https://books.google.co.uk/books?id=lLtyz275-GYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Dennis Washburn and Carole Cavanaugh (eds). Word and Image in Japanese Cinema. 2001. [https://books.google.co.uk/books?id=CwZ6SPpmJwQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Daisuke Miyao. The Aesthetics of Shadow: Lighting and Japanese Cinema. 2013. [https://books.google.co.uk/books?id=80K2AgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Justin Vicari. Japanese Film and the Floating Mind: Cinematic Contemplations of Being. 2016. [https://books.google.co.uk/books?id=CYuwDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Keiko I McDonald. Cinema East: A Critical Study of Major Japanese Films. Fairleigh Dickinson University Press. Associated University Presses. 1983. [https://books.google.co.uk/books?id=ywxjAAAAMAAJ]
*Alastair Phillips and Julian Stringer. Japanese Cinema: Texts and Contexts. 2007. [https://books.google.co.uk/books?id=53xRa5dwUOUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darrell William Davis. Picturing Japaneseness: Monumental Style, National Identity, Japanese Film. Columbia University Press. 1996. ISBN 0-231-10231-3. Review: [https://www.cambridge.org/core/journals/journal-of-asian-studies/article/abs/picturing-japaneseness-monumental-style-national-identity-japanese-film-by-darrell-william-davis-new-york-columbia-university-press-1996-viii-304-pp-1750/7CC73AD51C9A06CA40DF3463CC9D3E7D]

Bibliography
*Nornes and Gerow. &quot;Annotated Bibliography for Bibliographic Studies&quot;. Research Guide to Japanese Film Studies. Chapter 4. [https://books.google.co.uk/books?id=5UONCwAAQBAJ&amp;pg=PA72#v=onepage&amp;q&amp;f=false p 72].

Reference
*[https://guides.library.yale.edu/c.php?g=295932&amp;p=1973050 Japanese Reference Materials for Studying Japanese Cinema at Yale University]. Yale Library.

Periodicals

See also [[w:ja:Category:日本の映画雑誌]]. [映画雑誌 = film magazine] [映画機関誌 = film journal]
*Honchi Haruhiko (Japanese: 本地陽彦). Nihon Eiga Zasshi Taitoru Soran (Japanese: 日本映画雑誌タイトル総覧). [[w:ja:ワイズ出版|Waizu Shuppan]]. Tokyo. 2003. [https://books.google.co.uk/books?id=kskvAQAAIAAJ]. Commentary: Research Guide to Japanese Film Studies, [https://books.google.co.uk/books?id=5UONCwAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false p 83].
*[https://www.nfaj.go.jp/exhibition/filmmagazines/ 映画雑誌の秘かな愉しみ The Discreet Charm of Film Magazines]. National Film Archive of Japan (NFAJ).
*[https://guides.library.yale.edu/c.php?g=295932&amp;p=1973061 Japanese Reference Materials for Studying Japanese Cinema at Yale University: Film Periodicals]. Yale Library.
*[[w:en:Kinema Junpo|Kinema Junpo]] (Japanese: キネマ旬報) (The Movie Times) [https://books.google.co.uk/books?id=UFKtYBwc5ioC]
*[[w:en:UniJapan Film Quarterly|UniJapan Film Quarterly]] [https://books.google.co.uk/books?id=QIY3AAAAIAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/AA12490478]

Annuals and year books
*Japanese Films. UniJapan Film. [https://books.google.co.uk/books?id=4LAaAQAAIAAJ 1960]
**Japanese Film [https://books.google.co.uk/books?id=iJUHAQAAIAAJ 1983] [https://books.google.co.uk/books?id=fpYHAQAAIAAJ]
*Cinema Year Book of Japan. [https://books.google.co.uk/books?id=r7W9DZnBbBwC 1938].

History
*Jasper Sharp. Historical Dictionary of Japanese Cinema. 2011. [https://books.google.co.uk/books?id=YQR4EQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Isolde Standish. A New History of Japanese Cinema: A Century of Narrative Film. 2006. [https://books.google.co.uk/books?id=GZxUv4g_icQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Donald Ritchie. A Hundred Years of Japanese Film: A Concise History, with a Selective Guide to DVDs and Videos. 2001. 2005. [https://books.google.co.uk/books?id=s7-_Gon5-a0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yomota Inuhiko. What Is Japanese Cinema? A History. 2014. 2019. [https://books.google.co.uk/books?id=8Il-DwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Gerald Mast and Bruce F Kawin. &quot;Japan&quot; in &quot;Cinemas East&quot;. A Short History of the Movies. Allyn and Bacon. 7th Ed: 2000: [https://books.google.co.uk/books?id=AC0IAQAAMAAJ]. pp 403 to 417.
*Aaron Gerow. &quot;From Misemono to Zigomar: A Discursive History of Early Japanese Cinema&quot;. Bean, Kapse and Horak (eds). Silent Cinema and the Politics of Space. 2014. Chapter 6. [https://books.google.co.uk/books?id=7poiAwAAQBAJ&amp;pg=PA157#v=onepage&amp;q&amp;f=false p 157].

20th century
*Beverley Bare Buehrer. Japanese Films: A Filmography and Commentary, 1921-1989. McFarland &amp; Company. 1990. [https://books.google.co.uk/books?id=N6RZAAAAMAAJ]
*Aristides Gazetas. &quot;Post-War Japanese Cinema: 1950-1990&quot;. An Introduction to World Cinema. 2nd Ed. 2008. Chapter 13. [https://books.google.co.uk/books?id=CPuZ-2UtVRwC&amp;pg=PA169#v=onepage&amp;q&amp;f=false p 169].
*Japanese Experimental Film, 1960-1980. American Federation of Arts. [https://books.google.co.uk/books?id=XGUbAQAAIAAJ]
*Yuriko Furuhata. Cinema of Actuality: Japanese Avant-Garde Filmmaking in the Season of Image Politics. 2013. [https://books.google.co.uk/books?id=sDi2AgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [covers 1960s and 1970s].
*Outlaw Masters of Japanese Film. 2005. [https://books.google.co.uk/books?id=MbaKDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [covers 1950s to 1970s]
*Japanese Cinema: From Kurosawa To Tora-San&quot; (1986) Information Bulletin, February 1986, p 15 [https://books.google.co.uk/books?id=Sw5PAQAAIAAJ]
*Catherine Russell. Classical Japanese Cinema Revisited. 2011. [https://books.google.co.uk/books?id=uv9GAQAAQBAJ&amp;pg=PR4#v=onepage&amp;q&amp;f=false] [covers roughly 1930 to 1960]
*Kyoko Hirano. Mr. Smith Goes to Tokyo: Japanese Cinema Under the American Occupation, 1945-1952. 1992. [https://books.google.co.uk/books?id=6OsKAQAAMAAJ]
*Peter B High. The Imperial Screen: Japanese Film Culture in the Fifteen Years' War, 1931-1945. 2003. [https://books.google.co.uk/books?id=6XiA9DOuvjAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Naoki Yamamoto. Dialectics without Synthesis: Japanese Film Theory and Realism in a Global Frame. 2020. [https://books.google.co.uk/books?id=lEnrDwAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false] [Covers 1910s to 1950s].
*Sean O'Reilly (ed). The Advent of Sound in Japanese Cinema: A Handbook. 2025. [https://books.google.co.uk/books?id=vAKbEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [covers the 1930s]
*Mitsuyo Wada-Marciano. Nippon Modern: Japanese Cinema of the 1920s and 1930s. 2008. [https://books.google.co.uk/books?id=dIc0RBYMs9kC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Iwamoto Kenji, &quot;Japanese Cinema Until 1930: A Consideration of its Formal Aspects&quot;. Iris: A Journal of Theory on Image and Sound. No 16, p 9 [https://books.google.co.uk/books?id=zXFhCc7ENAEC]
*David Bordwell, &quot;Visual Style in Japanese Cinema, 1925-1945&quot; (1995) [https://books.google.co.uk/books?id=YqIqAQAAIAAJ 7] Film History 5 to 31 (No 1: Spring 1995)
*Sean D O'Reilly. Re-Viewing the Past: The Uses of History in the Cinema of Imperial Japan. 2018. Paperback Ed: 2020. [https://books.google.co.uk/books?id=fqFiDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

New, contemporary
*Mark Schilling. Contemporary Japanese Film. 1999. [https://books.google.co.uk/books?id=LNQ4EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kawamoto Saburo. New Trends in Japanese Cinema. (Orientation Seminars on Japan, No 21). The Japan Foundation, Office for the Japanese Studies Center. [https://books.google.co.uk/books?id=R5IqAQAAIAAJ]
*Tadao Sato. Currents in Japanese Cinema. Kodansha International. 1982. [https://books.google.com/books?id=p-9kAAAAMAAJ]

Genres
*Abe Mark Nornes. Japanese Documentary Film: The Meiji Era Through Hiroshima. 2003. [https://books.google.co.uk/books?id=2JFrwF2LMw4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*S A Thornton. The Japanese Period Film: A Critical Analysis. 2008. [https://books.google.co.uk/books?id=yhPo95wPz8QC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Alain Silver. The Samurai Film. 1977: [https://books.google.co.uk/books?id=R4BZAAAAMAAJ]. The Overlook Press. 1983: [https://books.google.co.uk/books?id=AH1ZAAAAMAAJ]
*David Desser. The Samurai Films of Akira Kurosawa. UMI Research Press. [https://books.google.com/books?id=AaJZAAAAMAAJ]
*Silver Screen Samurai: The Best of Japan's Samurai Movie Posters. 2004. [https://books.google.co.uk/books?id=tdsuoae7X68C&amp;pg=PA2#v=onepage&amp;q&amp;f=false]
*Stuart Galbraith IV. Japanese Science Fiction, Fantasy and Horror Films. Macfarland. 1994. [https://books.google.co.uk/books?id=2XZREQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Thomas Weisser and Yuko Mihara Weisser. Japanese Cinema Encyclopedia: The Horror, Fantasy, and SciFi films. Vital Books. 1997. [https://books.google.com/books?id=HGSybwAACAAJ]
*Jay McRoy. Japanese Horror Cinema. University of Hawaii Press. Edinburgh University Press. 2005. [https://books.google.co.uk/books?id=XvOgEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] Review: Timothy Iles (2007) [https://books.google.co.uk/books?id=rt9KAQAAIAAJ 33] The Journal of Japanese Studies 264
*Colette Balmain. Introduction to Japanese Horror Film. 2008. [https://books.google.co.uk/books?id=POWqBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Michael Crandol. Ghost in the Well: The Hidden History of Horror Films in Japan. 2021. [https://books.google.co.uk/books?id=mDgfEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Peter C Pugsley. Japanese High School Films: Iconography, Nostalgia and Discipline. 2022. [https://books.google.co.uk/books?id=jPOgEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Peter C Pugsley. Beyond the High School Film: Reaching Adulthood in Everyday Japanese Cinema. 2026. [https://books.google.co.uk/books?id=7boEEgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cult
*Patrick Macias. Tokyoscope: The Japanese Cult Film Companion. 2001. [https://books.google.co.uk/books?id=g-0HAQAAMAAJ]

Directors
*Alexander Jacoby. A Critical Handbook of Japanese Film Directors: From the Silent Era to the Present Day. 2008. [https://books.google.co.uk/books?id=RhWNAgAAQBAJ&amp;pg=PA1870#v=onepage&amp;q&amp;f=false]
*Audie Bock. Japanese Film Directors. 1978. Paperback Ed. Kodansha International. 1985. [https://books.google.co.uk/books?id=A6EqAAAAYAAJ]
*Mitsuhiro Yoshimoto. Kurosawa: Film Studies and Japanese Cinema. 2000. [https://books.google.co.uk/books?id=QizaCOjKs-IC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Stars
*Hideaki Fujiki. Making Personas: Transnational Film Stardom in Modern Japan. 2013. [https://books.google.co.uk/books?id=A_gFEAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

Studios
*Stuart Galbraith IV. The Toho Studios Story: A History and Complete Filmography. 2008. [https://books.google.co.uk/books?id=f7o8pq6G_dYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Interviews
*Joan Mellen. Voices from the Japanese Cinema. Liveright. New York. 1975. ISBN 0871406047.

V-Cinema
*Tom Mes. Japanese Film and the Challenge of Video. 2023. [https://books.google.co.uk/books?id=70i7EAAAQBAJ&amp;pg=PA1921#v=onepage&amp;q&amp;f=false]

Film criticism
*Scott Nygren. Time Frames: Japanese Cinema And the Unfolding of History. [https://books.google.co.uk/books?id=PvlgQgAACAAJ]

[[Category:Film]]</text>
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  <page>
    <title>User talk:Samuel Abraham Steinberg</title>
    <ns>3</ns>
    <id>331718</id>
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      <id>2829712</id>
      <timestamp>2026-08-30T12:06:21Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Econofoods */ new section</comment>
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      <text bytes="538" sha1="4rma8e0e9p589yhd7yon7156rz6bldo" xml:space="preserve">== [[Econofoods]] ==

Hi Samuel!

I've unfortunately had to delete your recent page as it seemed encyclopedic. Encyclopedia articles should be hosted on Wikipedia. Please see [[Wikiversity:What is Wikiversity?]], [[Wikiversity:What Wikiversity is not]], and [[Wikiversity:Learning by doing]] to get a better understanding of what content is appropriate. Please let me know if you have any questions or concerns,

—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:06, 30 August 2026 (UTC)</text>
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  <page>
    <title>User talk:Xazyaqan</title>
    <ns>3</ns>
    <id>331719</id>
    <revision>
      <id>2829713</id>
      <timestamp>2026-08-30T12:07:00Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Welcome */ new section</comment>
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      <text bytes="2313" sha1="9dslmbccer9y4hndjmhlmwobk5trh0a" xml:space="preserve">==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]],  Xazyaqan!'''|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Atcovi|me personally]] if you would like some [[Help:Contents|help]].

Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.

We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies.

To find your way around, check out:
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Wikiversity:Introduction|Introduction to Wikiversity]]
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* [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;

* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]]
* Learn [[Help:How to write an educational resource|how to write an educational resource]]
* Find out about [[Wikiversity:Research|research]] activities
* Give [[Wikiversity:Feedback|feedback]] about your observations
* Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].

See you around Wikiversity! --—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:07, 30 August 2026 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
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  <page>
    <title>Talk:Problems in living</title>
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      <timestamp>2026-08-30T18:20:48Z</timestamp>
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      <comment>Created page with &quot;can the deleted content be restored and copied/moved to my user space so I can see what was here? thanks and limitless peace. ~~~~&quot;</comment>
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      <text bytes="273" sha1="38rycdo7sqqs33ss0rniambv244lvrd" xml:space="preserve">can the deleted content be restored and copied/moved to my user space so I can see what was here? thanks and limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 18:20, 30 August 2026 (UTC)</text>
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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
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      <comment>/* */ Reply</comment>
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      <text bytes="565" sha1="7vvacmkqd7wgmrf03d8uh7gacytxhuh" xml:space="preserve">can the deleted content be restored and copied/moved to my user space so I can see what was here? thanks and limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 18:20, 30 August 2026 (UTC)

:{{ping|Michael Ten}} Hi Michael. Are you sure you meant this page? Because there is literally no intelligible content that was removed. Check the deletion reasons. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:26, 30 August 2026 (UTC)</text>
      <sha1>7vvacmkqd7wgmrf03d8uh7gacytxhuh</sha1>
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    <revision>
      <id>2829752</id>
      <parentid>2829751</parentid>
      <timestamp>2026-08-30T18:28:27Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
      </contributor>
      <comment>/* */ Reply</comment>
      <origin>2829752</origin>
      <model>wikitext</model>
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      <text bytes="859" sha1="n41qfs0ingnspr7c71v60q1sa74apsu" xml:space="preserve">can the deleted content be restored and copied/moved to my user space so I can see what was here? thanks and limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 18:20, 30 August 2026 (UTC)

:{{ping|Michael Ten}} Hi Michael. Are you sure you meant this page? Because there is literally no intelligible content that was removed. Check the deletion reasons. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:26, 30 August 2026 (UTC)
::i did mean this. i was not sure if deletion reason was maybe erroneous. nevermind if it is correct and really is blank.... thanks and bless up. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 18:28, 30 August 2026 (UTC)</text>
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  <page>
    <title>Problems in living</title>
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      <timestamp>2026-08-30T18:55:55Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
      </contributor>
      <comment>Created page with &quot;Problems in living are problems that human encounter, and they can be minor or major in severity. Problems in living can cause inconvenience. Problems in living can elicit emotional distress. There are many ways to potentially solve problems in living. Some problems in living plague humanity for hundreds or thousands of years. Some problems in living are inherent in the human condition. Regardless, humans have flourished for thousands of years, and if one looks back at h...&quot;</comment>
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      <text bytes="3667" sha1="nq1e3c987di24zdmrovydz0vhv6it7l" xml:space="preserve">Problems in living are problems that human encounter, and they can be minor or major in severity. Problems in living can cause inconvenience. Problems in living can elicit emotional distress. There are many ways to potentially solve problems in living. Some problems in living plague humanity for hundreds or thousands of years. Some problems in living are inherent in the human condition. Regardless, humans have flourished for thousands of years, and if one looks back at humanity even longer, for tens and arguably hundreds of thousands of years, since humans evolved from other hominid species millennia ago. [[File:The Thinker (24284172271).jpg|thumb|right]]

This area is partially to present problems in living from a Szaszian perspective. 

According to psychiatrist Thomas Szasz, psychiatry is the medicalization of problems in living. Thomas Szasz was a psychiatry profession and wrote extensively about the history of psychiatry and in relation to that also, humans existential and practical need to solve problems in living in life. 

==Discussion questions==

* What might society look like if most all problems in living are solved for all of humanity? 
* Is it generally better to prioritize solving major or minor problems in living? 
** What types of problems in living are best to solve first?
* Are there other ways that problems in living can be categorized or conceptualized? 
* What are the unintended consequences of medicalizing problems in living? 
* Throughout the history of education, how has research been applied to help humans solve and resolve problems in living? 
* What are good strategies for solving problems in living? 
* What role does personal responsibility play in identifying and solving problems in living? 

== Categories of problems in living ==
[[Economic problems]]
[[Social problems]]
[[Spiritual problems]]
[[Ecological problems]]
[[Political problems]]
[[Health problems]]
[[Philosophical problems]]
[[Technological problems]]

== Specific examples of problems and examples ===
* People in a geographic region do not have sufficient access to caloric needs. 
* Balancing the need for power for AI data centers with the needs of the entire population. 
* Expensive prices of high bandwidth memory. 
* Lack of clean drinking water. 
* No back up plan for humanity in case another huge meteor impacts Earth like during the time of the dinosaurs. 
* Fuel prices being too expensive. 
* Social conflict between family members. 
* Social conflict between friends. 
* Someone feeling a lack of meaning and purpose in life. 
* A lack of educational opportunities. 
* Feeling stuck in a job that does not pay a living wage. 
* A lack of reliable transportation. 
* There can be an infinite # of types of problems in living. 

== Strategies for solving problems in living ==
* Apply [[academic research]] to problems in living. 
* Start a [[business]] around helping other solve problems in living. 
* Identify problems in living. Prioritize which problems in living will benefit most from solving first. Create a plan to solve the problem in living. Implement the plan to solve the problem in living. Evaluate the success of the implemented plans. If the plan was successful, choose a new problem in living to solve and repeat these steps. If the plan was not successful, change the failed plan or create a new plan to solve the problem in living and try to implement and then evaluate the new or modified plan for solving the problem in living. 

== See also ==
* [[Psychiatry]]
* [[Psychology]]
* [[Thomas Szasz]]
* [[Social factors influencing health]] / [[Social determinants of health]]
* [[Sociology]]
* [[Anthropology]]</text>
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      <text bytes="3683" sha1="6mnthroiclm6qk9kk86g895qid6zr1n" xml:space="preserve">Problems in living are problems that human encounter, and they can be minor or major in severity. Problems in living can cause inconvenience. Problems in living can elicit emotional distress. There are many ways to potentially solve problems in living. Some problems in living plague humanity for hundreds or thousands of years. Some problems in living are inherent in the human condition. Regardless, humans have flourished for thousands of years, and if one looks back at humanity even longer, for tens and arguably hundreds of thousands of years, since humans evolved from other hominid species millennia ago. [[File:The Thinker (24284172271).jpg|thumb|right]]

This area is partially to present problems in living from a Szaszian perspective. 

According to psychiatrist Thomas Szasz, psychiatry is the medicalization of problems in living. Thomas Szasz was a psychiatry profession and wrote extensively about the history of psychiatry and in relation to that also, humans existential and practical need to solve problems in living in life. 

==Discussion questions==

* What might society look like if most all problems in living are solved for all of humanity? 
* Is it generally better to prioritize solving major or minor problems in living? 
** What types of problems in living are best to solve first?
* Are there other ways that problems in living can be categorized or conceptualized? 
* What are the unintended consequences of medicalizing problems in living? 
* Throughout the history of education, how has research been applied to help humans solve and resolve problems in living? 
* What are good strategies for solving problems in living? 
* What role does personal responsibility play in identifying and solving problems in living? 

== Categories of problems in living ==
* [[Economic problems]]
* [[Social problems]]
* [[Spiritual problems]]
* [[Ecological problems]]
* [[Political problems]]
* [[Health problems]]
* [[Philosophical problems]]
* [[Technological problems]]

== Specific examples of problems and examples ===
* People in a geographic region do not have sufficient access to caloric needs. 
* Balancing the need for power for AI data centers with the needs of the entire population. 
* Expensive prices of high bandwidth memory. 
* Lack of clean drinking water. 
* No back up plan for humanity in case another huge meteor impacts Earth like during the time of the dinosaurs. 
* Fuel prices being too expensive. 
* Social conflict between family members. 
* Social conflict between friends. 
* Someone feeling a lack of meaning and purpose in life. 
* A lack of educational opportunities. 
* Feeling stuck in a job that does not pay a living wage. 
* A lack of reliable transportation. 
* There can be an infinite # of types of problems in living. 

== Strategies for solving problems in living ==
* Apply [[academic research]] to problems in living. 
* Start a [[business]] around helping other solve problems in living. 
* Identify problems in living. Prioritize which problems in living will benefit most from solving first. Create a plan to solve the problem in living. Implement the plan to solve the problem in living. Evaluate the success of the implemented plans. If the plan was successful, choose a new problem in living to solve and repeat these steps. If the plan was not successful, change the failed plan or create a new plan to solve the problem in living and try to implement and then evaluate the new or modified plan for solving the problem in living. 

== See also ==
* [[Psychiatry]]
* [[Psychology]]
* [[Thomas Szasz]]
* [[Social factors influencing health]] / [[Social determinants of health]]
* [[Sociology]]
* [[Anthropology]]</text>
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      <text bytes="3681" sha1="kqd8mvm651ioaly4h8g84o2yxlpsbgi" xml:space="preserve">Problems in living are problems that human encounter, and they can be minor or major in severity. Problems in living can cause inconvenience. Problems in living can elicit emotional distress. There are many ways to potentially solve problems in living. Some problems in living plague humanity for hundreds or thousands of years. Some problems in living are inherent in the human condition. Regardless, humans have flourished for thousands of years, and if one looks back at humanity even longer, for tens and arguably hundreds of thousands of years, since humans evolved from other hominid species millennia ago. [[File:The Thinker (24284172271).jpg|thumb|right]]

This area is partially to present problems in living from a Szaszian perspective. 

According to psychiatrist Thomas Szasz, psychiatry is the medicalization of problems in living. Thomas Szasz was a psychiatry profession and wrote extensively about the history of psychiatry and in relation to that also, humans existential and practical need to solve problems in living in life. 

==Discussion questions==

* What might society look like if most all problems in living are solved for all of humanity? 
* Is it generally better to prioritize solving major or minor problems in living? 
** What types of problems in living are best to solve first?
* Are there other ways that problems in living can be categorized or conceptualized? 
* What are the unintended consequences of medicalizing problems in living? 
* Throughout the history of education, how has research been applied to help humans solve and resolve problems in living? 
* What are good strategies for solving problems in living? 
* What role does personal responsibility play in identifying and solving problems in living? 

== Categories of problems in living ==
* [[Economic problems]]
* [[Social problems]]
* [[Spiritual problems]]
* [[Ecological problems]]
* [[Political problems]]
* [[Health problems]]
* [[Philosophical problems]]
* [[Technological problems]]

== Specific examples of problems and examples ==
* People in a geographic region do not have sufficient access to caloric needs. 
* Balancing the need for power for AI data centers with the needs of the entire population. 
* Expensive prices of high bandwidth memory. 
* Lack of clean drinking water. 
* No back up plan for humanity in case another huge meteor impacts Earth like during the time of the dinosaurs. 
* Fuel prices being too expensive. 
* Social conflict between family members. 
* Social conflict between friends. 
* Someone feeling a lack of meaning and purpose in life. 
* A lack of educational opportunities. 
* Feeling stuck in a job that does not pay a living wage. 
* A lack of reliable transportation. 
* There can be an infinite # of types of problems in living.

== Strategies for solving problems in living ==
* Apply [[academic research]] to problems in living. 
* Start a [[business]] around helping other solve problems in living. 
* Identify problems in living. Prioritize which problems in living will benefit most from solving first. Create a plan to solve the problem in living. Implement the plan to solve the problem in living. Evaluate the success of the implemented plans. If the plan was successful, choose a new problem in living to solve and repeat these steps. If the plan was not successful, change the failed plan or create a new plan to solve the problem in living and try to implement and then evaluate the new or modified plan for solving the problem in living. 

== See also ==
* [[Psychiatry]]
* [[Psychology]]
* [[Thomas Szasz]]
* [[Social factors influencing health]] / [[Social determinants of health]]
* [[Sociology]]
* [[Anthropology]]</text>
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      <text bytes="3687" sha1="gindnnl0y4iivtbit6vnbc5yz25anaj" xml:space="preserve">'''Problems in living''' are problems that human encounter, and they can be minor or major in severity. Problems in living can cause inconvenience. Problems in living can elicit emotional distress. There are many ways to potentially solve problems in living. Some problems in living plague humanity for hundreds or thousands of years. Some problems in living are inherent in the human condition. Regardless, humans have flourished for thousands of years, and if one looks back at humanity even longer, for tens and arguably hundreds of thousands of years, since humans evolved from other hominid species millennia ago. [[File:The Thinker (24284172271).jpg|thumb|right]]

This area is partially to present problems in living from a Szaszian perspective. 

According to psychiatrist Thomas Szasz, psychiatry is the medicalization of problems in living. Thomas Szasz was a psychiatry profession and wrote extensively about the history of psychiatry and in relation to that also, humans existential and practical need to solve problems in living in life. 

==Discussion questions==

* What might society look like if most all problems in living are solved for all of humanity? 
* Is it generally better to prioritize solving major or minor problems in living? 
** What types of problems in living are best to solve first?
* Are there other ways that problems in living can be categorized or conceptualized? 
* What are the unintended consequences of medicalizing problems in living? 
* Throughout the history of education, how has research been applied to help humans solve and resolve problems in living? 
* What are good strategies for solving problems in living? 
* What role does personal responsibility play in identifying and solving problems in living? 

== Categories of problems in living ==
* [[Economic problems]]
* [[Social problems]]
* [[Spiritual problems]]
* [[Ecological problems]]
* [[Political problems]]
* [[Health problems]]
* [[Philosophical problems]]
* [[Technological problems]]

== Specific examples of problems and examples ==
* People in a geographic region do not have sufficient access to caloric needs. 
* Balancing the need for power for AI data centers with the needs of the entire population. 
* Expensive prices of high bandwidth memory. 
* Lack of clean drinking water. 
* No back up plan for humanity in case another huge meteor impacts Earth like during the time of the dinosaurs. 
* Fuel prices being too expensive. 
* Social conflict between family members. 
* Social conflict between friends. 
* Someone feeling a lack of meaning and purpose in life. 
* A lack of educational opportunities. 
* Feeling stuck in a job that does not pay a living wage. 
* A lack of reliable transportation. 
* There can be an infinite # of types of problems in living.

== Strategies for solving problems in living ==
* Apply [[academic research]] to problems in living. 
* Start a [[business]] around helping other solve problems in living. 
* Identify problems in living. Prioritize which problems in living will benefit most from solving first. Create a plan to solve the problem in living. Implement the plan to solve the problem in living. Evaluate the success of the implemented plans. If the plan was successful, choose a new problem in living to solve and repeat these steps. If the plan was not successful, change the failed plan or create a new plan to solve the problem in living and try to implement and then evaluate the new or modified plan for solving the problem in living. 

== See also ==
* [[Psychiatry]]
* [[Psychology]]
* [[Thomas Szasz]]
* [[Social factors influencing health]] / [[Social determinants of health]]
* [[Sociology]]
* [[Anthropology]]</text>
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      <text bytes="4005" sha1="qu838x1drpvzwx1vrnzi9gmx22la6fj" xml:space="preserve">'''Problems in living''' are problems that human encounter, and they can be minor or major in severity. This area is to help facilitate an understanding of problems in living as it relates to what they are, how they can be identified and conceptualized, and how learning, teaching, and research can be utilized to help solve and resolve problems in living so that all of humanity can benefit, individually and collectively. Problems in living can cause inconvenience. Problems in living can elicit emotional distress. There are many ways to potentially solve problems in living. Some problems in living plague humanity for hundreds or thousands of years. Some problems in living are inherent in the human condition. Regardless, humans have flourished for thousands of years, and if one looks back at humanity even longer, for tens and arguably hundreds of thousands of years, since humans evolved from other hominid species millennia ago. [[File:The Thinker (24284172271).jpg|thumb|right]]

This area is partially to present problems in living from a Szaszian perspective. According to psychiatrist Thomas Szasz, psychiatry is the medicalization of problems in living. Thomas Szasz was a psychiatry profession and wrote extensively about the history of psychiatry and in relation to that also, humans existential and practical need to solve problems in living in life. 

==Discussion questions==

* What might society look like if most all problems in living are solved for all of humanity? 
* Is it generally better to prioritize solving major or minor problems in living? 
** What types of problems in living are best to solve first?
* Are there other ways that problems in living can be categorized or conceptualized? 
* What are the unintended consequences of medicalizing problems in living? 
* Throughout the history of education, how has research been applied to help humans solve and resolve problems in living? 
* What are good strategies for solving problems in living? 
* What role does personal responsibility play in identifying and solving problems in living? 

== Categories of problems in living ==
* [[Economic problems]]
* [[Social problems]]
* [[Spiritual problems]]
* [[Ecological problems]]
* [[Political problems]]
* [[Health problems]]
* [[Philosophical problems]]
* [[Technological problems]]

== Specific examples of problems and examples ==
* People in a geographic region do not have sufficient access to caloric needs. 
* Balancing the need for power for AI data centers with the needs of the entire population. 
* Expensive prices of high bandwidth memory. 
* Lack of clean drinking water. 
* No back up plan for humanity in case another huge meteor impacts Earth like during the time of the dinosaurs. 
* Fuel prices being too expensive. 
* Social conflict between family members. 
* Social conflict between friends. 
* Someone feeling a lack of meaning and purpose in life. 
* A lack of educational opportunities. 
* Feeling stuck in a job that does not pay a living wage. 
* A lack of reliable transportation. 
* There can be an infinite # of types of problems in living.

== Strategies for solving problems in living ==
* Apply [[academic research]] to problems in living. 
* Start a [[business]] around helping other solve problems in living. 
* Identify problems in living. Prioritize which problems in living will benefit most from solving first. Create a plan to solve the problem in living. Implement the plan to solve the problem in living. Evaluate the success of the implemented plans. If the plan was successful, choose a new problem in living to solve and repeat these steps. If the plan was not successful, change the failed plan or create a new plan to solve the problem in living and try to implement and then evaluate the new or modified plan for solving the problem in living. 

== See also ==
* [[Psychiatry]]
* [[Psychology]]
* [[Thomas Szasz]]
* [[Social factors influencing health]] / [[Social determinants of health]]
* [[Sociology]]
* [[Anthropology]]</text>
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      <comment>added categories</comment>
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      <text bytes="4192" sha1="ac7v2sap5q7ozu7kzj3xry2pz1atmbd" xml:space="preserve">'''Problems in living''' are problems that human encounter, and they can be minor or major in severity. This area is to help facilitate an understanding of problems in living as it relates to what they are, how they can be identified and conceptualized, and how learning, teaching, and research can be utilized to help solve and resolve problems in living so that all of humanity can benefit, individually and collectively. Problems in living can cause inconvenience. Problems in living can elicit emotional distress. There are many ways to potentially solve problems in living. Some problems in living plague humanity for hundreds or thousands of years. Some problems in living are inherent in the human condition. Regardless, humans have flourished for thousands of years, and if one looks back at humanity even longer, for tens and arguably hundreds of thousands of years, since humans evolved from other hominid species millennia ago. [[File:The Thinker (24284172271).jpg|thumb|right]]

This area is partially to present problems in living from a Szaszian perspective. According to psychiatrist Thomas Szasz, psychiatry is the medicalization of problems in living. Thomas Szasz was a psychiatry profession and wrote extensively about the history of psychiatry and in relation to that also, humans existential and practical need to solve problems in living in life. 

==Discussion questions==

* What might society look like if most all problems in living are solved for all of humanity? 
* Is it generally better to prioritize solving major or minor problems in living? 
** What types of problems in living are best to solve first?
* Are there other ways that problems in living can be categorized or conceptualized? 
* What are the unintended consequences of medicalizing problems in living? 
* Throughout the history of education, how has research been applied to help humans solve and resolve problems in living? 
* What are good strategies for solving problems in living? 
* What role does personal responsibility play in identifying and solving problems in living? 

== Categories of problems in living ==
* [[Economic problems]]
* [[Social problems]]
* [[Spiritual problems]]
* [[Ecological problems]]
* [[Political problems]]
* [[Health problems]]
* [[Philosophical problems]]
* [[Technological problems]]

== Specific examples of problems and examples ==
* People in a geographic region do not have sufficient access to caloric needs. 
* Balancing the need for power for AI data centers with the needs of the entire population. 
* Expensive prices of high bandwidth memory. 
* Lack of clean drinking water. 
* No back up plan for humanity in case another huge meteor impacts Earth like during the time of the dinosaurs. 
* Fuel prices being too expensive. 
* Social conflict between family members. 
* Social conflict between friends. 
* Someone feeling a lack of meaning and purpose in life. 
* A lack of educational opportunities. 
* Feeling stuck in a job that does not pay a living wage. 
* A lack of reliable transportation. 
* There can be an infinite # of types of problems in living.

== Strategies for solving problems in living ==
* Apply [[academic research]] to problems in living. 
* Start a [[business]] around helping other solve problems in living. 
* Identify problems in living. Prioritize which problems in living will benefit most from solving first. Create a plan to solve the problem in living. Implement the plan to solve the problem in living. Evaluate the success of the implemented plans. If the plan was successful, choose a new problem in living to solve and repeat these steps. If the plan was not successful, change the failed plan or create a new plan to solve the problem in living and try to implement and then evaluate the new or modified plan for solving the problem in living. 

== See also ==
* [[Psychiatry]]
* [[Psychology]]
* [[Thomas Szasz]]
* [[Social factors influencing health]] / [[Social determinants of health]]
* [[Sociology]]
* [[Anthropology]]

[[Category:Philosophy of psychiatry]]
[[Category:Social problems]]
[[Category:Problem solving]]
[[Category:Social determinants of health]]
[[Category:Psychology]]
[[Category:Sociology]]</text>
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  <page>
    <title>User:Michael Ten/Futurism</title>
    <ns>2</ns>
    <id>331724</id>
    <revision>
      <id>2829760</id>
      <timestamp>2026-08-30T18:58:09Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
      </contributor>
      <comment>creating copy to edit  - original contributors seen here - https://en.wikiversity.org/w/index.php?title=User:Marshallsumter/Futurism&amp;action=history  LIMITLESS PEACE</comment>
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      <text bytes="6627" sha1="7cz7nmi997iz75gte2ue1vso4z4h6jm" xml:space="preserve">[[Image:Feed Your Mind Futurism.png|thumb|right|200px|This is an artist's impression of futurism or feed your mind, brain, futurism, future studies, futurology. Credit: [[User:Ps2045|Singularity Utopia]].]]
'''Futurism''' in a general sense is concern with events and trends of the future or which anticipate the future. It was also an art movement begun in Italy in 1909 to celebrate and incorporate the energy and dynamism of modern technology. The art movement apparently ended in 1918. Perhaps it was interrupted by World War I.
{{RightTOC}}

==Theoretical futurism==
'''Def.''' a &quot;study and prediction of possible futures&quot;&lt;ref name=FuturismWikt&gt;{{ cite web
|author=[[wikt:User:Imran~enwiktionary|Imran~enwiktionary]]
|title=futurism, In: ''Wiktionary''
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=1 September 2015
|url=https://en.wiktionary.org/wiki/futurism
|accessdate=2015-09-17 }}&lt;/ref&gt; is called '''futurism'''.

==The future of futurism==

&quot;By combining the best of three different approaches to futurism--crisis futurism, evolutionary futurism, and spiritual futurism--we can realize vast human potentials and, perhaps, even attain the next stage of human evolution.&quot;&lt;ref name=Hubbard&gt;{{ cite journal
|author=Barbara Marx Hubbard
|title=The Future of Futurism: Creating a New Synthesis
|journal=Futurist
|month=April
|year=1983
|volume=17
|issue=2
|pages=52-8
|url=http://eric.ed.gov/?id=EJ278746
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Before hominins had large brains they first started walking upright. To evolve larger brains we need to be taller with appropriate support structure or we break our necks ever more often. To evolve the ability to fly we need to become smaller, have a lot of holes in our bones and fit our brains into a much smaller space. Or, we could invent aircraft and spacecraft and enjoy more recreational activity. There are always bills to pay.

&quot;The effort to think systematically about the future began little more than a half-century ago, and the results so far have not been impressive. Today's futurists hope that more sophisticated methods will allow them to provide a better picture of what tomorrow may bring.&quot;&lt;ref name=Rejeski&gt;{{ cite journal
|author=David Rejeski and Robert L. Olson
|title=Has Futurism Failed?
|journal=The Wilson Quarterly
|month=Winter
|year=2006
|volume=30
|issue=1
|pages=14-21
|url=http://www.jstor.org/stable/40261340
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2015-09-17 }}&lt;/ref&gt;

Part of the problem with accurately predicting or trying to predict the future is that hedging and investment specialists are more likely to try to own the future and its inventors and inventions to maximize their gain, especially the more accurate the predictions become.

==Readings and learning media==
===Wikipedia===
{{colbegin|3}}
* [[Wikipedia: Technological singularity|Technological singularity]]
* [[Wikipedia: Futures studies|Futures studies]]
* [[Wikipedia: Systems thinking|Systems thinking]]
* [[Wikipedia: Technology roadmap|Technology roadmap]]
* [[Wikipedia: Future workshop|Future workshop]]
* [[Wikipedia: Technology forecasting|Technology forecasting]]
* [[Wikipedia: Trend analysis|Trend analysis]]
* [[Wikipedia: Futurist|Futurist]]
* [[Wikipedia: Decentralized autonomous organization|Decentralized autonomous organization]]
* [[Wikipedia: InterPlanetary File System|InterPlanetary File System]]
* [[Wikipedia: Accelerating change|Accelerating change]]
* [[Wikipedia: List of emerging technologies|List of emerging technologies]]
* [[Wikipedia: Outline of futures studies|Outline of futures studies]]
* [[Wikipedia: Singularity University|Singularity University]] - How can [[business]] and futurism intersect? 
* [[Wikipedia: Artificial uterus|Artificial uterus]] - a potential future technology that could eliminate physical suffering from child birth and reduce [[political science|political conflicts]]. 
* [[Wikipedia: NASA Institute for Advanced Concepts|NASA Institute for Advanced Concepts]]
* [[Wikipedia: Ledger (journal)|Ledger (journal)]] - Peer-reviewed academic journal covering research on all aspects of cryptocurrencies and blockchain technology, including mathematics, computer science, engineering, law, economics and philosophy.
* [[w:List of hypothetical technologies|List of hypothetical technologies]]
* [[w:Post-scarcity economy|Post-scarcity economy]]
* [[w:The End of Work|The End of Work]]
* [[w:Post-work society|Post-work society]]
{{colend}}

==Discussion questions==
* How can a better understanding of futurism and futurology help to better and benefit humanity and society? 
* How do futurism, [[business]], and [[economics]] intersect? What research could [[WikiJournal of Business and Economics|be disseminated]] in relation to such intersections?
* How can futurism be utilized to help mitigate [[environmental problems]]?
* How can futurism be utilized to help address [[social problems]] in a successful way?

==Areas of interest/sub niches==
{{colbegin|3}}
* [[Cryonics]]
* [[Life extension]]
* [[Singularity]]
* [[Exponential technologies]]
* [[Strategies for Engineered Negligible Senescence]]
* [[Basic income]]
* [[Overcoming pro-aging mindsets]]
* [[Anthropomorphic robotics]]
* [[Crowd funding]]
* [[Wikipedia:Cultured meat]]
* [[Vertical farming]]
* [[Renewable energy]]
* [[Artificial intelligence]]
* [[Exponential technologies]]
* [[Asteroid mining]]
* [[3D printing]]
* [[High-temperature superconductivity]]
* [[Colonizing Mars]]
* [[Off Earth colonization]]
* [[Nanotechnology]]
* [[Technological automation]]
* [[Unmanned aerial systems|Drones]]
* [[Decentralized autonomous organization|Decentralized autonomous organizations]]
* [[Synthetic biology]]
* [[Quantum computing]]
* [[Envisioning Our Future]]
* [[Robotics]]
* [[Rejuvenation]]
* [[Blockchain]] 
* [[Cryptocurrencies]] [[Wikipedia:Cryptocurrency]]
* [[Cryptoeconomics]]
* [[Post scarcity]]
* [[Fusion energy]]
* [[Supercomputing]]/[[Supercomputers]]
{{colend}}

==See also==
* [[Technological singularity]]
* [[Introduction to Futures Studies]]

==External links==
{{Sisterprojectsearch}}
{{article}}
* [https://hpluspedia.org H+Pedia] - large transhumanist and futurist wiki from Humanity+
* [http://futurology.reddit.com/ Futurology on Reddit]
* [https://www.fhi.ox.ac.uk/ Future of Humanity Institute at Oxford University]
* [https://su.org/ Singularity University]
* [http://cryptoubi.reddit.com Discuss universal basic income implemented through cryptocurrencies on Reddit] or start a conversation about it at [[Cryptocurrency basic income]]

==References==
{{reflist}}</text>
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      <comment>Topic development feedback</comment>
      <origin>2829788</origin>
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      <format>text/x-wiki</format>
      <text bytes="4664" sha1="3jcbaj7sssgkjpnxquq5dv1w9x0z7cd" xml:space="preserve">== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:|Hi [[User:{{{1}}}|{{{1}}}]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:31, 30 August 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
&lt;!-- 1-level --&gt;
# Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings)
# There appears to be a second-level of heading that hasn't been styled as such (see Tutorial 2 for a how-to)
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
# Consider shortening or splitting the scenario
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
# Consider abbreviating, using plain English, and moving most or all citations into subsequent sections
&lt;!-- Focus questions --&gt;
# Focus questions are aligned with sub-title and top-level headings
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Excellent – key points are well developed for each section
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
&lt;!-- Theory and research --&gt;
# Promising balance of theory and research
&lt;!-- Citations --&gt;
# Excellent use of citations
&lt;!-- Conclusion --&gt;
# Conclusion is well underway
# As with the Overview, use plain English, moving most of the citations into earlier sections, and concentrating this section on the practical, take-home messages? (i.e., address each focus questions)
|5=
&lt;!-- Figure --&gt;
# Relevant figure(s) are presented and captioned
&lt;!-- Cite --&gt;
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
&lt;!-- Scenarios/examples/case studies --&gt;
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
# Excellent use of quiz question(s)
&lt;!-- Tables --&gt;
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Basic
&lt;!-- Systematic reviews --&gt;
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## alphabetical order
## capitalisation
## [[Help:Wikitext quick reference|italicisation]]
## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting]
## Remove &quot;doi: &quot;
## page numbers should be separated by an en-dash (–) rather than a hyphen (-)
|8=
&lt;!-- Resources --&gt;
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# See also
### Also link to relevant [[w:|Wikipedia]] pages
## Use alphabetical order
&lt;!-- External links --&gt;
# External links
## OK
## Only include links directly related to the sub-title
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
|9=
&lt;!-- User page --&gt;
# Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
|10=
&lt;!-- Social contribution --&gt;
# None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]].
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:31, 30 August 2026 (UTC)</text>
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      <comment>Created page with &quot;{{DISPLAYTITLE:Motivation and emotion/Book/2026/Deliberative vs implemental mindset u3252175 - u3252175}}  == '''Deliberative vs Implemental Mindset''' == The deliberative and implemental mindset are two distinct cognitive processes associated with differing stages in the pursuit of a goal. The distinction of these mindsets was developed and coined by psychologist Peter Gollwitzer during his work on the Rubicon of Action theory and later superseded by the Mindsets in the...&quot;</comment>
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== '''Deliberative vs Implemental Mindset''' ==
The deliberative and implemental mindset are two distinct cognitive processes associated with differing stages in the pursuit of a goal. The distinction of these mindsets was developed and coined by psychologist Peter Gollwitzer during his work on the Rubicon of Action theory and later superseded by the Mindsets in the Control of Action theory developed later on. The deliberative mindset is primarily associated with deciding which goal to pursue, whilst the implemental mindset is associated with the how and how to achieve proposed goal.

== '''Background''' ==
According to Gollwitzer, goal directed action involved four distinct stages, including deciding what they as an individual want to achieve, deciding how an individual will achieve said goal and assessing the outcome of achieving proposed goal. Gollwitzer proposed that differing stages involved in this process created differing psychological demands and activated different cognitive processes, or mindsets, to meet these demands. The distinction Gollwitzer made originated from researching the difference between pre-decisional and post-decisional stages in which an individual is made to consider their goal, it's desirability and feasibility, as it considers that desires and needs to be beyond what is achievable in a lifetime. After a decision has been made attention or mindset will shift towards the necessary steps need to achieve said goal, or planning and implementation.

=== Mindset of Theory of Action ===
The mindset of theory of action is separated into four stages that can be associated with both the deliberative and implementation activated across the four stages of goal pursuit

'''Pre-decisional phase''' '''-''' deliberating on a goal, weighing up feasibility and desirability

'''Post-decisional phase''' '''-''' implementation of action on a goal

'''Actional Phase''' '''-''' carrying out sustained action to achieve said goal

'''Post- Actional phase''' '''-''' assessing out outcome of desired and reached goal

The mindset of action perfectly demonstrates the deliberative and implementation mindset in action when undertaking goal directed behaviour.

=== '''Deliberative Mindset''' ===
A deliberative mindset occurs during the pre-decisional stage of pursuing a goal, an individual is weighing up desirability and feasibility and have not committed themselves to a goal to pursue.

==== '''Characteristics of a deliberative mindset''' ====

* Consideration of multiple options, pathways and alternatives
* Consider desirability and feasibility of proposed goals
* Be realistic of likelihood of success of goal
* Take consideration of available information in support or contradiction of goal

A deliberative mindset can be summed in a nutshell when considering this scenario, an individual is freshly eighteen years old, they have completed their HSC, so what's next? Do they pursue vocational study, do a gap year, pursue full-time employment or pursue university.

In this scenario, an individual is entering the pre-decisional phase or activating their deliberative mindset, weighing up cost,duration, benefits, potential career pathways and likelihood of success which is itself based on aptitude and inclination. So the question is, what's next? 

=== '''Implementation Mindset''' ===
What's next occurs after an individual has committed to their pursuit of a goal and begins planning how pursue that goal and the necessary steps needed to achieve that goal. Therefore it is associated with the post decisional stage of goal pursuit. The individual has moved on and deliberated on the feasibility and the likelihood of success of the proposed goal. The individual will then focus cognitive resources on how to achieve said goal. So the next question is, how do i achieve my goal?

==== '''Characteristics of an Implementation mindset''' ====

* Focus on the when, why, where and how
* Identify the actions needed to achieve settled on goal
* Anticipate and develop strategies to overcome obstacles and potential difficulties


Consider the earlier scenario, an individual is freshly eighteen, they have completed their HSC, so the pathways become vocational study (apprenticeship), university, gap year or full-time employment. They've identified that they would pursue psychology, so what's next for them? They decide which university, weighing potential cost, reputation, they determine study/life to achieve the best grades without burning out, where psychology will take them, will it be human resources, marketing, recruiting or healthcare. Psychology can be applied anywhere it is an incredibly broad discipline.

=== Comparison ===
{| class=&quot;wikitable&quot;
! colspan=&quot;undefined&quot; |Feature
! colspan=&quot;undefined&quot; |Deliberative Mindset
! colspan=&quot;undefined&quot; |Implemental Mindset
|-
| colspan=&quot;undefined&quot; |'''Action Phase'''
| colspan=&quot;undefined&quot; |Pre-decisional (Goal Setting)
| colspan=&quot;undefined&quot; |Post-decisional (Goal Striving)
|-
| colspan=&quot;undefined&quot; |'''Cognitive Style'''
| colspan=&quot;undefined&quot; |Open-minded, impartial
| colspan=&quot;undefined&quot; |Closed-minded, partial
|-
| colspan=&quot;undefined&quot; |'''Information Focus'''
| colspan=&quot;undefined&quot; |Feasibility and desirability pros/cons
| colspan=&quot;undefined&quot; |Execution steps and &quot;how-to&quot; plans
|-
| colspan=&quot;undefined&quot; |'''Attention Span'''
| colspan=&quot;undefined&quot; |Broad visual and mental attention
| colspan=&quot;undefined&quot; |Narrowed, focused on foreground tasks
|}

=== Limitations ===
The distinction between the deliberative and implementation mindset exhibits that in pursuit of your own individual goals, differentiating mindsets are essential in the pursuit of a goal. In a deliberative mindset, an individual considers their options, the steps that it&lt;nowiki&gt;''&lt;/nowiki&gt;ll take to reach goal, whereas an implementation mindset will activate when a decision has been reached and requires sustained long-term action.

However, it's important to consider an excessively long sustained deliberative mindset may lead to indecisiveness, procrastination and difficulty taking action. Whilst, an implementation mindset elicited too hastily may lead to an ill-informed choice or an undesirable or unfeasible outcome. It should be consider that each individual is different and mindsets change as new information is received either substantially increasing desirability and feasibility or having the opposite effect. 

An individual may decide on the university pathway, as opposed to vocational study but lose out on long-term financial gains, if they were to do an apprenticeship or traineeship, they would have to consider job security and whether technological advancement can affect their future career.

== '''References''' ==


Keller, L., Gollwitzer, P., &amp; Sheeran, P. (n.d.). ''6 Changing Behavior Using the Model of Action Phases''. Retrieved August 31, 2026, from https://bpb-us-e1.wpmucdn.com/wp.nyu.edu/dist/c/6235/files/2020/08/Keller-Gollwitzer-Sheeran-Changing_behavior_using_the_model_of_action_phases.pdf?bid=6235   

Gollwitzer, P. M., &amp; Bayer, U. (2026). ''Deliberative versus implemental mindsets in the control of action | Social Psychology &amp; Motivation''. Uni-Konstanz.De. https://www.socmot.uni-konstanz.de/publications/deliberative-versus-implemental-mindsets-control-action?utm_

Brandstatter, V. (2015, March). ''(PDF) the role of deliberative versus implemental mindsets in time prediction and task accomplishment''. ResearchGate. https://www.researchgate.net/publication/275632731_The_Role_of_Deliberative_Versus_Implemental_Mindsets_in_Time_Prediction_and_Task_Accomplishment

Keller, L., Bieleke, M., &amp; Gollwitzer, P. M. (2019). Mindset theory of action phases and IF-then planning. ''Social Psychology in Action'', 23–37. https://doi.org/10.1007/978-3-030-13788-5_2


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== '''Deliberative vs Implemental Mindset''' ==
The deliberative and implemental mindset are two distinct cognitive processes associated with differing stages in the pursuit of a goal. The distinction of these mindsets was developed and coined by psychologist Peter Gollwitzer during his work on the Rubicon of Action theory and later superseded by the Mindsets in the Control of Action theory developed later on. The deliberative mindset is primarily associated with deciding which goal to pursue, whilst the implemental mindset is associated with the how and how to achieve proposed goal.

== '''Background''' ==
According to Gollwitzer, goal directed action involved four distinct stages, including deciding what they as an individual want to achieve, deciding how an individual will achieve said goal and assessing the outcome of achieving proposed goal. Gollwitzer proposed that differing stages involved in this process created differing psychological demands and activated different cognitive processes, or mindsets, to meet these demands. The distinction Gollwitzer made originated from researching the difference between pre-decisional and post-decisional stages in which an individual is made to consider their goal, it's desirability and feasibility, as it considers that desires and needs to be beyond what is achievable in a lifetime. After a decision has been made attention or mindset will shift towards the necessary steps need to achieve said goal, or planning and implementation.

=== Mindset of Theory of Action ===
The mindset of theory of action is separated into four stages that can be associated with both the deliberative and implementation activated across the four stages of goal pursuit

'''Pre-decisional phase''' '''-''' deliberating on a goal, weighing up feasibility and desirability

'''Post-decisional phase''' '''-''' implementation of action on a goal

'''Actional Phase''' '''-''' carrying out sustained action to achieve said goal

'''Post- Actional phase''' '''-''' assessing out outcome of desired and reached goal

The mindset of action perfectly demonstrates the deliberative and implementation mindset in action when undertaking goal directed behaviour.

=== '''Deliberative Mindset''' ===
A deliberative mindset occurs during the pre-decisional stage of pursuing a goal, an individual is weighing up desirability and feasibility and have not committed themselves to a goal to pursue.

==== '''Characteristics of a deliberative mindset''' ====

* Consideration of multiple options, pathways and alternatives
* Consider desirability and feasibility of proposed goals
* Be realistic of likelihood of success of goal
* Take consideration of available information in support or contradiction of goal

A deliberative mindset can be summed in a nutshell when considering this scenario, an individual is freshly eighteen years old, they have completed their HSC, so what's next? Do they pursue vocational study, do a gap year, pursue full-time employment or pursue university.

In this scenario, an individual is entering the pre-decisional phase or activating their deliberative mindset, weighing up cost,duration, benefits, potential career pathways and likelihood of success which is itself based on aptitude and inclination. So the question is, what's next? 

=== '''Implementation Mindset''' ===
What's next occurs after an individual has committed to their pursuit of a goal and begins planning how pursue that goal and the necessary steps needed to achieve that goal. Therefore it is associated with the post decisional stage of goal pursuit. The individual has moved on and deliberated on the feasibility and the likelihood of success of the proposed goal. The individual will then focus cognitive resources on how to achieve said goal. So the next question is, how do i achieve my goal?

==== '''Characteristics of an Implementation mindset''' ====

* Focus on the when, why, where and how
* Identify the actions needed to achieve settled on goal
* Anticipate and develop strategies to overcome obstacles and potential difficulties


Consider the earlier scenario, an individual is freshly eighteen, they have completed their HSC, so the pathways become vocational study (apprenticeship), university, gap year or full-time employment. They've identified that they would pursue psychology, so what's next for them? They decide which university, weighing potential cost, reputation, they determine study/life to achieve the best grades without burning out, where psychology will take them, will it be human resources, marketing, recruiting or healthcare. Psychology can be applied anywhere it is an incredibly broad discipline.

=== Comparison ===
{| class=&quot;wikitable&quot;
! colspan=&quot;undefined&quot; |Feature
! colspan=&quot;undefined&quot; |Deliberative Mindset
! colspan=&quot;undefined&quot; |Implemental Mindset
|-
| colspan=&quot;undefined&quot; |'''Action Phase'''
| colspan=&quot;undefined&quot; |Pre-decisional (Goal Setting)
| colspan=&quot;undefined&quot; |Post-decisional (Goal Striving)
|-
| colspan=&quot;undefined&quot; |'''Cognitive Style'''
| colspan=&quot;undefined&quot; |Open-minded, impartial
| colspan=&quot;undefined&quot; |Closed-minded, partial
|-
| colspan=&quot;undefined&quot; |'''Information Focus'''
| colspan=&quot;undefined&quot; |Feasibility and desirability pros/cons
| colspan=&quot;undefined&quot; |Execution steps and &quot;how-to&quot; plans
|-
| colspan=&quot;undefined&quot; |'''Attention Span'''
| colspan=&quot;undefined&quot; |Broad visual and mental attention
| colspan=&quot;undefined&quot; |Narrowed, focused on foreground tasks
|}

=== Limitations ===
The distinction between the deliberative and implementation mindset exhibits that in pursuit of your own individual goals, differentiating mindsets are essential in the pursuit of a goal. In a deliberative mindset, an individual considers their options, the steps that it&lt;nowiki&gt;''&lt;/nowiki&gt;ll take to reach goal, whereas an implementation mindset will activate when a decision has been reached and requires sustained long-term action.

However, it's important to consider an excessively long sustained deliberative mindset may lead to indecisiveness, procrastination and difficulty taking action. Whilst, an implementation mindset elicited too hastily may lead to an ill-informed choice or an undesirable or unfeasible outcome. It should be consider that each individual is different and mindsets change as new information is received either substantially increasing desirability and feasibility or having the opposite effect. 

An individual may decide on the university pathway, as opposed to vocational study but lose out on long-term financial gains, if they were to do an apprenticeship or traineeship, they would have to consider job security and whether technological advancement can affect their future career.

== '''References''' ==


Keller, L., Gollwitzer, P., &amp; Sheeran, P. (n.d.). ''6 Changing Behavior Using the Model of Action Phases''. Retrieved August 31, 2026, from https://bpb-us-e1.wpmucdn.com/wp.nyu.edu/dist/c/6235/files/2020/08/Keller-Gollwitzer-Sheeran-Changing_behavior_using_the_model_of_action_phases.pdf?bid=6235   

Gollwitzer, P. M., &amp; Bayer, U. (2026). ''Deliberative versus implemental mindsets in the control of action | Social Psychology &amp; Motivation''. Uni-Konstanz.De. https://www.socmot.uni-konstanz.de/publications/deliberative-versus-implemental-mindsets-control-action?utm_

Brandstatter, V. (2015, March). ''(PDF) the role of deliberative versus implemental mindsets in time prediction and task accomplishment''. ResearchGate. https://www.researchgate.net/publication/275632731_The_Role_of_Deliberative_Versus_Implemental_Mindsets_in_Time_Prediction_and_Task_Accomplishment

Keller, L., Bieleke, M., &amp; Gollwitzer, P. M. (2019). Mindset theory of action phases and IF-then planning. ''Social Psychology in Action'', 23–37. https://doi.org/10.1007/978-3-030-13788-5_2

=== Gen Ai Statement ===
I acknowledge usage of ai in completion of this assignment, its usage was utilised in determining structure, finding sources and assisted with content but with thorough fact checking and re-write into my own words upon my understanding of the subject matter.


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      <comment>Created page with &quot;= Overview = As defined by specialists, Artificial Intelligences (AI), such as ChatGPT, are online tools that are trained by using Natural Language Processing (NLP) to create responses. During this process, the AI is given input from words or phrases to create vectors that represent their meaning. The AI systems use algorithms that are trained based on social data to process the information provided.&lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://ifttt.com/explore/how-does-chat-gp...&quot;</comment>
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      <text bytes="5506" sha1="ni74lsffir1ndxb6d0ohcvoj8hcsivt" xml:space="preserve">= Overview =
As defined by specialists, Artificial Intelligences (AI), such as ChatGPT, are online tools that are trained by using Natural Language Processing (NLP) to create responses. During this process, the AI is given input from words or phrases to create vectors that represent their meaning. The AI systems use algorithms that are trained based on social data to process the information provided.&lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://ifttt.com/explore/how-does-chat-gpt-work|title=How does ChatGPT work?|last=The IFTTT Team|date=2024, August 2|year=2024|website=IFTTT}}&lt;/ref&gt; Conversely, other experts claim that AI also implicates the access to biased information: the most popular information, but not necessarily good.&lt;ref name=&quot;:1&quot;&gt;{{Cite book|title=Artificial unintelligence: How computers misunderstand the world|last=Broussard|first=Meredith|publisher=The MIT Press|year=2018|isbn=9780262346726|location=Cambridge, United Kingdom (England)|pages=146-157}}&lt;/ref&gt;

=== &lt;small&gt;'''AI Functioning and Uses'''&lt;/small&gt; ===
Artificial Intelligences are online tools that are trained by using Natural Language Processing to generate human-like responses based on different prompts.&lt;ref name=&quot;:0&quot; /&gt; These systems are trained through supervised learning on large amounts of data, allowing them to identify patterns in language and generate responses that are appropriate to the user's input. According to current research, the tool contains a self-attention layer that focuses on specific words or phrases to provide specific responses. For this reason, it is usually said that the program is designed to work similarly to the human brain. This pre-trained algorithm uses vectors to represent the meaning of the word in a mathematical form. Furthermore, the system can be refined through human feedback, which helps improve the quality and relevance of its responses over time. Due to these characteristics, AI can be used in different contexts and for a variety of purposes.&lt;ref name=&quot;:0&quot; /&gt; It is becoming increasingly integral to daily and professional life due to its capacity to automate tasks. Firstly, it allows for creating social media content and writing assistance. Secondly, it provides help for programming and technical solutions. Finally, it can assist students and educators in their professional development. 

=== '''AI limitations''' ===
Many scholars believe that AI algorithms are not independent entities since they are designed by people and inevitably reflect human assumptions and biases. The data that is collected and processed by the algorithm is not neutral or purely objective, rather it is socially constructed.&lt;ref name=&quot;:1&quot; /&gt; Since computers are products of human design, the biases that exist in the real world are also reproduced in computational systems.&lt;ref&gt;{{Cite web|url=https://issues.org/algorithm-auditing-more-than-glitch-broussard/|title=How to Investigate an Algorithm|last=Broussard|first=Meredith|date=2023|website=Issues in Science and Technology 39, no. 4}}&lt;/ref&gt; Other authors explain that algorithms learn from “deep machine learning” to replicate human thinking which is predicated on specific values from different kinds of people. Therefore, these professionals claim that the access to biased data reinforces oppressive social relationships.&lt;ref name=&quot;:2&quot;&gt;{{Cite book|title=Algorithms of oppression: How search engines reinforce racism|last=Noble|first=Safiya Umoja|publisher=New York University Press|year=2018|isbn=9781479833641|pages=1-14}}&lt;/ref&gt;

=== '''Data bias in Artificial Intelligence''' ===
AI systems are trained on large amounts of social data. During training, the model processes this information using a pre-trained transformer algorithm, which relies on a self-attention mechanism to identify relevant words or phrases and generate responses. The input is divided into smaller units and represented mathematically through vectors, allowing the system to recognize patterns within the data.&lt;ref name=&quot;:0&quot; /&gt; According to researchers, these datasets may contain racial and gender biases that can contribute to the consolidation of existing social and economic inequalities.&lt;ref name=&quot;:2&quot; /&gt; Since all data is generated, collected or organized by people, it is not naturally objective or neutral. Rather, it is socially constructed because it reflects the historical, cultural and social contexts in which it is produced. As a result, datasets may contain omissions, inconsistencies and biases that become part of the information used to train AI systems. The information that becomes more visible in digital environments is often the most popular because algorithms prioritize engagement and visibility. However, popularity does not indicate quality or accuracy.&lt;ref name=&quot;:1&quot; /&gt; All in all, AI algorithms may reproduce existing social inequalities rather than operate as neutral systems.

=== '''Relying on AI''' ===
Other experts on the topic maintain that machines can only operate within the limits established by the programming, as they simply execute instructions.&lt;ref name=&quot;:1&quot; /&gt; Consequently, these systems offer limitations, as they cannot provide solutions to all possible or complex social situations. Their operation depends on human intervention, computational resources and energy infrastructure. Human beings, by contrast, are the only ones who possess the capacity to exercise critical thinking, ethical judgment and contextual reasoning when facing complex situations. 

== '''References''' ==
&lt;references /&gt;</text>
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    <revision>
      <id>2829822</id>
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      <contributor>
        <username>Sol Doporto</username>
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      <text bytes="5528" sha1="az02vefjmmf8g8w5frpyfwbilap5m2e" xml:space="preserve">= Overview =
As defined by specialists, Artificial Intelligences (AI), such as ChatGPT, are online tools that are trained by using Natural Language Processing (NLP) to create responses. During this process, the AI is given input from words or phrases to create vectors that represent their meaning. The AI systems use algorithms that are trained based on social data to process the information provided.&lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://ifttt.com/explore/how-does-chat-gpt-work|title=How does ChatGPT work?|last=The IFTTT Team|date=2024, August 2|year=2024|website=IFTTT|access-date=8/31/2026}}&lt;/ref&gt; Conversely, other experts claim that AI also implicates the access to biased information: the most popular information, but not necessarily good.&lt;ref name=&quot;:1&quot;&gt;{{Cite book|title=Artificial unintelligence: How computers misunderstand the world|last=Broussard|first=Meredith|publisher=The MIT Press|year=2018|isbn=9780262346726|location=Cambridge, United Kingdom (England)|pages=146-157}}&lt;/ref&gt;

=== &lt;small&gt;'''AI Functioning and Uses'''&lt;/small&gt; ===
Artificial Intelligences are online tools that are trained by using Natural Language Processing to generate human-like responses based on different prompts.&lt;ref name=&quot;:0&quot; /&gt; These systems are trained through supervised learning on large amounts of data, allowing them to identify patterns in language and generate responses that are appropriate to the user's input. According to current research, the tool contains a self-attention layer that focuses on specific words or phrases to provide specific responses. For this reason, it is usually said that the program is designed to work similarly to the human brain. This pre-trained algorithm uses vectors to represent the meaning of the word in a mathematical form. Furthermore, the system can be refined through human feedback, which helps improve the quality and relevance of its responses over time. Due to these characteristics, AI can be used in different contexts and for a variety of purposes.&lt;ref name=&quot;:0&quot; /&gt; It is becoming increasingly integral to daily and professional life due to its capacity to automate tasks. Firstly, it allows for creating social media content and writing assistance. Secondly, it provides help for programming and technical solutions. Finally, it can assist students and educators in their professional development. 

=== '''AI limitations''' ===
Many scholars believe that AI algorithms are not independent entities since they are designed by people and inevitably reflect human assumptions and biases. The data that is collected and processed by the algorithm is not neutral or purely objective, rather it is socially constructed.&lt;ref name=&quot;:1&quot; /&gt; Since computers are products of human design, the biases that exist in the real world are also reproduced in computational systems.&lt;ref&gt;{{Cite web|url=https://issues.org/algorithm-auditing-more-than-glitch-broussard/|title=How to Investigate an Algorithm|last=Broussard|first=Meredith|date=2023|website=Issues in Science and Technology 39, no. 4}}&lt;/ref&gt; Other authors explain that algorithms learn from “deep machine learning” to replicate human thinking which is predicated on specific values from different kinds of people. Therefore, these professionals claim that the access to biased data reinforces oppressive social relationships.&lt;ref name=&quot;:2&quot;&gt;{{Cite book|title=Algorithms of oppression: How search engines reinforce racism|last=Noble|first=Safiya Umoja|publisher=New York University Press|year=2018|isbn=9781479833641|pages=1-14}}&lt;/ref&gt;

=== '''Data bias in Artificial Intelligence''' ===
AI systems are trained on large amounts of social data. During training, the model processes this information using a pre-trained transformer algorithm, which relies on a self-attention mechanism to identify relevant words or phrases and generate responses. The input is divided into smaller units and represented mathematically through vectors, allowing the system to recognize patterns within the data.&lt;ref name=&quot;:0&quot; /&gt; According to researchers, these datasets may contain racial and gender biases that can contribute to the consolidation of existing social and economic inequalities.&lt;ref name=&quot;:2&quot; /&gt; Since all data is generated, collected or organized by people, it is not naturally objective or neutral. Rather, it is socially constructed because it reflects the historical, cultural and social contexts in which it is produced. As a result, datasets may contain omissions, inconsistencies and biases that become part of the information used to train AI systems. The information that becomes more visible in digital environments is often the most popular because algorithms prioritize engagement and visibility. However, popularity does not indicate quality or accuracy.&lt;ref name=&quot;:1&quot; /&gt; All in all, AI algorithms may reproduce existing social inequalities rather than operate as neutral systems.

=== '''Relying on AI''' ===
Other experts on the topic maintain that machines can only operate within the limits established by the programming, as they simply execute instructions.&lt;ref name=&quot;:1&quot; /&gt; Consequently, these systems offer limitations, as they cannot provide solutions to all possible or complex social situations. Their operation depends on human intervention, computational resources and energy infrastructure. Human beings, by contrast, are the only ones who possess the capacity to exercise critical thinking, ethical judgment and contextual reasoning when facing complex situations. 

== '''References''' ==
&lt;references /&gt;</text>
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    <title>User:JimKillock/Nos in Schola Latine loquimur/Capitulum Secundum On being late</title>
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      <comment>Created page with &quot;'''2. On Latecomers'''  ''Serius'' in general means the same as ''sero'': whoever comes after the appointed time comes late (''sero'' or ''serius''). But when some comparison is involved, ''serius'', not ''sero'', is used.  Whoever comes ''tarde'' (slowly) takes his time, and does not always truly come ''serius'' (too late).  '''Master.''' What is the meaning of this, Sempronius, that you are a whole half-hour late? How does this happen? Come, state your reason.  '''S.''...&quot;</comment>
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      <text bytes="4539" sha1="r27iug8d3tldzs0zgnu2zwdjlpswiww" xml:space="preserve">'''2. On Latecomers'''

''Serius'' in general means the same as ''sero'': whoever comes after the appointed time comes late (''sero'' or ''serius''). But when some comparison is involved, ''serius'', not ''sero'', is used.

Whoever comes ''tarde'' (slowly) takes his time, and does not always truly come ''serius'' (too late).

'''Master.''' What is the meaning of this, Sempronius, that you are a whole half-hour late? How does this happen? Come, state your reason.

'''S.''' It grieves me greatly, sir, that I was not here earlier, but one of my father's friends met me and kept questioning me about all sorts of things concerning our school, and I could not pass him by without failing in courtesy.

'''M.''' But you could have answered him briefly and yet politely. For now I accept your excuse, but see that another time you are here on time (''ad tempus, in tempore'').

And you, Valdemirus, came considerably late.

Where did you linger? Why did you delay? Why are you here so late?

'''V.''' I did come a little late, it is true, but not so long after you, sir.

'''M.''' Whether I am late is no concern of yours; but it is certainly your duty to be here on time. Yesterday too you came quite late, and it has even become a habit with you to come almost every day not merely a little, but much too late. I want you from now on to be here at the appointed hour. Do you understand?

'''V.''' Yes, entirely; and since the road to good conduct is never too late, I will strive with all my might and effort, sir, to mend this fault of mine.

'''M.''' We shall see. Be sure you keep your promise diligently; for sooner or later (''citius tardiusve'') the one who as a boy grows used to negligence will regret it. And you, Petronius, I would warn to beware; for you too were not here on time. I fear that you also may one day be sorry. Or will it not be too late afterward to cast off (''deponere, exuere'') a habit once deeply settled?

'''P.''' By your leave, sir, may I say that, after lying awake all night from a toothache, I got up a little late from bed this morning.

'''M.''' You excuse yourself too late. You clear yourself later than you should have. You have disclosed this to me after the fact; I do not accept an excuse whose truth is uncertain. Change your ways (''mend yourself''). You must be here on time.

Did not you too, Claudius, come a bit too late (''a little too late'')?

'''C.''' Only by a moment, sir (''by a moment of time, by a single instant—one minute''). For eight had just struck (''precisely'') when I entered the school.

'''M.''' So be it; but you too seem to have adopted the motto: «Late, but at some point all the same.»

'''C.''' Kindly hear me, I beg, sir. For Antonius is to blame, who, walking with me, kept saying over and over: «Don't be in such a hurry, Claudius; we'll be there in good time.» Had he not misled me, I would have come a quarter of an hour sooner and so in good time (there would have been no delay on my part at all).

'''M.''' Ought you to have been so gullible? Should you at once have believed what he had said? Ah—the Phrygians grow wise too late!

Now then, Antonius, that reminds me of another matter. Late last night, returning from the railway station, I saw you out in the street (''in the street''). What? Do you walk about alone like that into the depths of night?

'''A.''' By your leave, sir, I would say it is likely you are mistaken, since yesterday evening—of which my father will be witness—I went to bed at nine o'clock at the very latest.

'''M.''' It may be that I was mistaken; but I had no doubt at all that it was you.

'''A.''' It was not I, believe me, sir; for no one can be at Rome when he is at Athens. And besides, my father, who goes to bed last (''who is the last to retire''), locks up the house every day at ten in the evening. But I am used to getting into bed at eight at the earliest, or nine at the very latest.

'''M.''' Let us leave the matter undecided: later (''later'') we shall perhaps see whether I judged rightly or wrongly. You have already deceived me once with a lie; what happened afterward, none of those present is unaware.

You too, Edmundus, came far too late (''too little early'').

What excuse do you offer (''what excuse do you have'')?

'''E.''' I, sir?

'''M.''' Yes, you indeed.

'''E.''' Forgive me, I beg—since, as far as I know, I hardly came late. Someone was asking for me at the door at the very moment eight struck.

'''M.''' Could he not have come at another time?

'''E.''' He could not, sir,</text>
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      <text bytes="5717" sha1="qcqkok3ua9brsqaci3yha8hohyznjfh" xml:space="preserve">'''2. On Latecomers'''

''Serius'' in general means the same as ''sero'': whoever comes after the appointed time comes late (''sero'' or ''serius''). But when some comparison is involved, ''serius'', not ''sero'', is used.

Whoever comes ''tarde'' (slowly) takes his time, and does not always truly come ''serius'' (too late).

'''Master.''' What is the meaning of this, Sempronius, that you are a whole half-hour late? How does this happen? Come, state your reason.

'''S.''' It grieves me greatly, sir, that I was not here earlier, but one of my father's friends met me and kept questioning me about all sorts of things concerning our school, and I could not pass him by without failing in courtesy.

'''M.''' But you could have answered him briefly and yet politely. For now I accept your excuse, but see that another time you are here on time (''ad tempus, in tempore'').

And you, Valdemirus, came considerably late.

Where did you linger? Why did you delay? Why are you here so late?

'''V.''' I did come a little late, it is true, but not so long after you, sir.

'''M.''' Whether I am late is no concern of yours; but it is certainly your duty to be here on time. Yesterday too you came quite late, and it has even become a habit with you to come almost every day not merely a little, but much too late. I want you from now on to be here at the appointed hour. Do you understand?

'''V.''' Yes, entirely; and since the road to good conduct is never too late, I will strive with all my might and effort, sir, to mend this fault of mine.

'''M.''' We shall see. Be sure you keep your promise diligently; for sooner or later (''citius tardiusve'') the one who as a boy grows used to negligence will regret it. And you, Petronius, I would warn to beware; for you too were not here on time. I fear that you also may one day be sorry. Or will it not be too late afterward to cast off (''deponere, exuere'') a habit once deeply settled?

'''P.''' By your leave, sir, may I say that, after lying awake all night from a toothache, I got up a little late from bed this morning.

'''M.''' You excuse yourself too late. You clear yourself later than you should have. You have disclosed this to me after the fact; I do not accept an excuse whose truth is uncertain. Change your ways (''mend yourself''). You must be here on time.

Did not you too, Claudius, come a bit too late (''a little too late'')?

'''C.''' Only by a moment, sir (''by a moment of time, by a single instant—one minute''). For eight had just struck (''precisely'') when I entered the school.

'''M.''' So be it; but you too seem to have adopted the motto: «Late, but at some point all the same.»

'''C.''' Kindly hear me, I beg, sir. For Antonius is to blame, who, walking with me, kept saying over and over: «Don't be in such a hurry, Claudius; we'll be there in good time.» Had he not misled me, I would have come a quarter of an hour sooner and so in good time (there would have been no delay on my part at all).

'''M.''' Ought you to have been so gullible? Should you at once have believed what he had said? Ah—the Phrygians grow wise too late!

Now then, Antonius, that reminds me of another matter. Late last night, returning from the railway station, I saw you out in the street (''in the street''). What? Do you walk about alone like that into the depths of night?

'''A.''' By your leave, sir, I would say it is likely you are mistaken, since yesterday evening—of which my father will be witness—I went to bed at nine o'clock at the very latest.

'''M.''' It may be that I was mistaken; but I had no doubt at all that it was you.

'''A.''' It was not I, believe me, sir; for no one can be at Rome when he is at Athens. And besides, my father, who goes to bed last (''who is the last to retire''), locks up the house every day at ten in the evening. But I am used to getting into bed at eight at the earliest, or nine at the very latest.

'''M.''' Let us leave the matter undecided: later (''later'') we shall perhaps see whether I judged rightly or wrongly. You have already deceived me once with a lie; what happened afterward, none of those present is unaware.

You too, Edmundus, came far too late (''too little early'').

What excuse do you offer (''what excuse do you have'')?

'''E.''' I, sir?

'''M.''' Yes, you indeed.

'''E.''' Forgive me, I beg—since, as far as I know, I hardly came late. Someone was asking for me at the door at the very moment eight struck.

'''M.''' Could he not have come at another time?

'''E.''' He could not, sir, because, passing this way, he had a letter from my mother to deliver to me.

'''M.''' Go to your places. You latecomers are at fault that we have wasted so much time (''made such a loss of time'').

At any moment nine will strike, so that it is almost too late to begin class. Still, we may put a few little questions.

When did Horace die, Marcellus?

'''Mar.''' He died in the eleventh year before the birth of Christ.

'''M.''' You are mistaken: he died somewhat later (''later''). He died three years later (''triennio tardius''), in the eighth year before Christ. He was a little later in age than Cicero (''lived after him''), who died in the forty-fourth year before Christ.

When did Plautus live, Guilelmus (''live'')?

'''G.''' Plautus lived considerably after Cicero (''considerably later'').

'''M.''' Oho! Are you in your right mind? If he had lived later than Cicero, how would Cicero be praising his Latinity?

You are off by more than a whole century, since Plautus was born a hundred and twenty years before Cicero.

For the rest, all this you will learn accurately later (''later''), when we come to deal with Roman literature (''treat of Roman literature'').</text>
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  <page>
    <title>OpenStax College Success</title>
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      </contributor>
      <comment>Created page with &quot;See also [[OpenStax]]  == &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;College Success&lt;/big&gt; ==  == Summary == OpenStax ''College Success'' is a comprehensive and contemporary resource that serves First Year Experience, Student Success, and College Transition courses. Developed with the support of hundreds of faculty and coordinators, the book addresses the evolving challenges and opportunities of today’s diverse students. Engagement, self-analysis, personal responsibility, and student su...&quot;</comment>
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      <text bytes="816" sha1="pb9tp956yb3h2zxmp68bsyjy9a8aeum" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;College Success&lt;/big&gt; ==

== Summary ==
OpenStax ''College Success'' is a comprehensive and contemporary resource that serves First Year Experience, Student Success, and College Transition courses. Developed with the support of hundreds of faculty and coordinators, the book addresses the evolving challenges and opportunities of today’s diverse students. Engagement, self-analysis, personal responsibility, and student support are reflected throughout the material. ''College Success'' also includes an array of student surveys and opinion polls, and OpenStax will regularly provide the results to adopting faculty.  

* OpenStax College Success (original content). Available as pdf or web view.
* OpenStax College Success audiobook Available as audio textbook.</text>
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    <revision>
      <id>2829851</id>
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        <username>Andy?yes</username>
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      <text bytes="928" sha1="c6wnpsqnzl9bis4pf48hqg9u8zvb5nn" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;College Success&lt;/big&gt; ==

== Summary ==
OpenStax ''College Success'' is a comprehensive and contemporary resource that serves First Year Experience, Student Success, and College Transition courses. Developed with the support of hundreds of faculty and coordinators, the book addresses the evolving challenges and opportunities of today’s diverse students. Engagement, self-analysis, personal responsibility, and student support are reflected throughout the material. ''College Success'' also includes an array of student surveys and opinion polls, and OpenStax will regularly provide the results to adopting faculty.  

* [https://openstax.org/details/books/college-success OpenStax College Success] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/college-success/ OpenStax College Success audiobook] Available as audio textbook.</text>
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  <page>
    <title>OpenStax Entrepreneurship</title>
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    <revision>
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      <contributor>
        <username>Andy?yes</username>
        <id>3006471</id>
      </contributor>
      <comment>Created page with &quot;See also [[OpenStax]]  == &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;Entrepreneurship&lt;/big&gt; ==  == Summary == This textbook is intended for use in introductory Entrepreneurship classes at the undergraduate level. Due to the wide range of audiences and course approaches, the book is designed to be as flexible as possible. Theoretical and practical aspects are presented in a balanced manner, and specific components such as the business plan are provided in multiple formats. ''Entrepreneursh...&quot;</comment>
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      <text bytes="764" sha1="tja9a1d35jlxk3ki7fz9esdyhjt7w5l" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;Entrepreneurship&lt;/big&gt; ==

== Summary ==
This textbook is intended for use in introductory Entrepreneurship classes at the undergraduate level. Due to the wide range of audiences and course approaches, the book is designed to be as flexible as possible. Theoretical and practical aspects are presented in a balanced manner, and specific components such as the business plan are provided in multiple formats. ''Entrepreneurship'' aims to drive students toward active participation in entrepreneurial roles, and exposes them to a wide range of companies and scenarios.

* OpenStax Entrepreneurship (original content). Available as pdf or web view.
* OpenStax Entrepreneurship audiobook  Available as audio textbook.</text>
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      <format>text/x-wiki</format>
      <text bytes="878" sha1="jjdphnb3ljcryoo2fljh5vbrfm7vw6u" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;Entrepreneurship&lt;/big&gt; ==

== Summary ==
This textbook is intended for use in introductory Entrepreneurship classes at the undergraduate level. Due to the wide range of audiences and course approaches, the book is designed to be as flexible as possible. Theoretical and practical aspects are presented in a balanced manner, and specific components such as the business plan are provided in multiple formats. ''Entrepreneurship'' aims to drive students toward active participation in entrepreneurial roles, and exposes them to a wide range of companies and scenarios.

* [https://openstax.org/details/books/entrepreneurship OpenStax Entrepreneurship] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/entrepreneurship/ OpenStax Entrepreneurship audiobook]  Available as audio textbook.</text>
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    <title>Talk:Motivation and emotion/Book/2026/Oxytocin as a neuromodulator</title>
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      <timestamp>2026-08-31T09:56:22Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
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      <comment>Topic development feedback</comment>
      <origin>2829861</origin>
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      <text bytes="3960" sha1="iqdnair2452brx2y5t7dfw3myee580r" xml:space="preserve">&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
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&lt;!-- Headings --&gt;
# Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings)
# A promising, 2-level structure is planned
# See Tutorial 2 for how to create headings and subheadings
&lt;!-- Other ---&gt;
# Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections
# Insufficient alignment between sub-title, focus questions, and top-level headings
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
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|4=
&lt;!-- Key points--&gt;
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# Promising development
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title
&lt;!-- Theory and research --&gt;
# Balance theoretical content with critical synthesis of relevant research
&lt;!-- GenAI ---&gt;
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
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# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
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# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# Use and internal link style and rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Social contribution --&gt;
# One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
#* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]]
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:56, 31 August 2026 (UTC)</text>
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      <comment>added [[Category:Motivation and emotion/Book/Neuromodulators]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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      <comment>Created page with &quot;[[Category:Motivation and emotion/Book]]&quot;</comment>
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    <title>Classical Myth Quiz QuizPartialFileName</title>
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      <comment>Created page with &quot;== Overview == Welcome to the Classical Mythology Quiz! Test your knowledge of ancient Greek and Roman myths, gods, heroes, and legendary creatures.  == Quiz Questions ==  === Question 1 === Who is the Greek god of the sky and ruler of the Olympian gods? * ( ) Poseidon * ( ) Hades * (X) Zeus * ( ) Apollo  === Question 2 === Which hero completed twelve famous labors, including defeating the Nemean Lion? * ( ) Achilles * (X) Heracles (Hercules) * ( ) Perseus * ( ) Theseus...&quot;</comment>
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      <text bytes="884" sha1="cehng6sq638sm13mpeyl5jn2sncqfle" xml:space="preserve">== Overview ==
Welcome to the Classical Mythology Quiz! Test your knowledge of ancient Greek and Roman myths, gods, heroes, and legendary creatures.

== Quiz Questions ==

=== Question 1 ===
Who is the Greek god of the sky and ruler of the Olympian gods?
* ( ) Poseidon
* ( ) Hades
* (X) Zeus
* ( ) Apollo

=== Question 2 ===
Which hero completed twelve famous labors, including defeating the Nemean Lion?
* ( ) Achilles
* (X) Heracles (Hercules)
* ( ) Perseus
* ( ) Theseus

=== Question 3 ===
Who was cursed to hold up the celestial heavens for eternity after the Titanomachy?
* ( ) Prometheus
* (X) Atlas
* ( ) Cronus
* ( ) Epimetheus

=== Question 4 ===
What was the name of the monster with snakes for hair that turned people to stone when they looked at her?
* ( ) Scylla
* ( ) Charybdis
* (X) Medusa
* ( ) Sphinx

== See Also ==
* [[Classical Mythology]]
* [[Portal:Mythology]]</text>
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      <comment>Created page with &quot;* [[User:JimKillock/Nos in Schola Latine loquimur/Capitulum Secundum On being late]] * [[User:JimKillock/Nos in Schola Latine loquimur/Capitulum Tertium The manner of exusing oneself]]&quot;</comment>
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      <text bytes="184" sha1="bna1zzrmcedy9c4ung1vs0ye0qluos2" xml:space="preserve">* [[User:JimKillock/Nos in Schola Latine loquimur/Capitulum Secundum On being late]]
* [[User:JimKillock/Nos in Schola Latine loquimur/Capitulum Tertium The manner of exusing oneself]]</text>
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      <timestamp>2026-08-31T10:45:54Z</timestamp>
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      <comment>Created page with &quot;'''3. The Manner of Excusing Oneself'''  '''M.''' How, Alexius, is the verb ''to excuse'' construed?  '''A.''' Let me give an example, sir, to make the matter clear.  # Peter excuses himself and Paul before the teacher (or ''to'' the teacher). # Peter pleads ill health before the teacher, i.e. excuses himself on the ground of ill health (''excuses himself on''). # Peter excuses his delay (lateness) to the teacher, or excuses himself for his delay (''excuses himself for''...&quot;</comment>
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      <text bytes="5138" sha1="ii70numlcf5qrmvx800o8qncx45ij5e" xml:space="preserve">'''3. The Manner of Excusing Oneself'''

'''M.''' How, Alexius, is the verb ''to excuse'' construed?

'''A.''' Let me give an example, sir, to make the matter clear.

# Peter excuses himself and Paul before the teacher (or ''to'' the teacher).
# Peter pleads ill health before the teacher, i.e. excuses himself on the ground of ill health (''excuses himself on'').
# Peter excuses his delay (lateness) to the teacher, or excuses himself for his delay (''excuses himself for'').

'''M.''' If you were the teacher, what would you say to Peter?

'''A.''' You look for an excuse too often, Peter (''try to excuse yourself'').

The start of your speech always begins with an excuse.

You are always ready to invent or find a reason (''pretext'').

You readily plead that you are ill (''claim''); you always have reasons at hand.

More often than is right you are absent from school on the excuse of illness (on the pretext, under the name, by the feigning of illness; putting forward or alleging that you are ill; pleading, interposing or introducing as a reason that you are sick—''on the pretext'').

The excuse you have just used is not adequate (proper—''valid'').

That is an excuse you have only just thought up.

It is a fault that has nothing to excuse it (has no excuse).

You can have no excuse (''you are not excusable'').

What excuse can you bring?

You certainly have not satisfied me.

Nothing is easier for you than to make use of something as an excuse.

Your fault cannot be pardoned.

This is unworthy of pardon (lacks pardon—''inexcusable'').

That gives you no excuse (''does not excuse you'').

You defend your negligence with the excuse of ill health.

Do you think I am indulgent toward your faults (''to excuse'')?

Do not excuse your fault (''excuse yourself for'').

Do not clear the deed; take care not to clear yourself.

Yesterday you pleaded by way of excuse that your head ached; today you plead pains in the belly (gripes—''stomachache'').

You defend even your bad habit of coming late with the excuse of necessity. A thing of that sort is surely not excusable.

'''M.''' That is quite enough, Alexius. Now then, Fredericus, you too came late and have not yet excused yourself to me.

I want whoever comes late to excuse himself before me. Nor have you handed in your assignment to me. You have not cleared yourself with me over the neglected assignment.—

Good, your excuses carry weight with me (they seem to me just and legitimate).

And yours, Hilarius, I accept (I accept your satisfaction).

The rest likewise I hold excused.

Why, Valdemirus, have you not done your assignment?—What? Are you looking for an excuse (''you are trying to'')?

How do you dare plead weak eyesight (''weak sight'')?

You know how to gather up every ground of excuse (''excuse yourself in every way'').

You are well practiced in the use of the ill-health excuse.

What excuses have you?—I'm listening.

Be quiet, Petronius. I do not want anyone to cover another's fault with an excuse.

I do not want anyone to make excuses to me on another's behalf (''make another's excuses'').

Do you think you are more deserving of an excuse because you are a magistrate's son?

You ought rather to be ashamed—born of a respectable station and yet ignorant to the point of blame (''of an inexcusable ignorance'').

Do not be angry with me if I say frankly that I place not even the slightest trust in you.

See to it that you make up for such a lapse with tears rather than with words (''make amends''), otherwise I for my part will not pardon your excuses (''accept them'').

I too do not have eyes that carry their sight far (''I am short-sighted''), and yet I never fall short in my duty.

Anyone can plead a hindrance.

What do you want, Benedictus?

'''B.''' Eugenius asks, sir, that you hold him excused on account of illness (''sends his excuses'').

'''M.''' If he is really ill, then of course I free him from blame (clear him, release him).

You, Edmundus, did not finish the whole assignment. What do you offer as a reason (''in excuse'')?

'''E.''' I forgot to finish what I had left off.

'''M.''' That cannot serve as an excuse for you.

'''E.''' By your leave, sir, I would say I did it unintentionally. I beg you to be willing to hold me excused (''to kindly excuse me''). My aunt (''amita'', father's sister; ''matertera'', mother's sister—''aunt'') had invited me to dinner, which I could not decline (''I could not refuse to accept'').

'''M.''' I pardon your excuses (I accept them). It is time to end class. You may go.—

'''E.''' Hurrah, class is dismissed! Today I shall dine at my uncle's, who has invited you too, Carolus.

'''C.''' Please make my excuses to your uncle (''give him my excuses''—I want to be excused to your uncle; may your uncle hold me excused), for—and consider this said for you alone (''this is for you only'')—when I recently accepted his invitation (''to accept an invitation''), he was apologizing to his guests over a meager little supper because the cook had been struck down by a serious illness. I will not let my expectations be disappointed a second time.</text>
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    <title>User:JonAwbrey/Figures and Tables 54</title>
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      <timestamp>2026-08-31T11:06:07Z</timestamp>
      <contributor>
        <username>JonAwbrey</username>
        <id>29537</id>
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      <comment>add user work page</comment>
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      <text bytes="15955" sha1="se736hcwxwnj10x16g4fv49jl44tpe5" xml:space="preserve">==Syntax and Semantics of a Calculus for Propositional Logic==

===PNG===

Table 1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives.

{| align=&quot;center&quot; style=&quot;font-size:large; text-align:center&quot;
|+ height=&quot;30px&quot; | &lt;math&gt;\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}&lt;/math&gt;
|-
| [[File:Syntax and Semantics of a Calculus for Propositional Logic 4.0.png|600px]]
|}

===Wiki + LaTeX + JPG===

Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable &lt;math&gt;k&lt;/math&gt;-ary scope.

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; |
&lt;math&gt;\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Graph}&lt;/math&gt;
| &lt;math&gt;\text{Expression}&lt;/math&gt;
| &lt;math&gt;\text{Interpretation}&lt;/math&gt;
| &lt;math&gt;\text{Other Notations}&lt;/math&gt;
|-
| height=&quot;100px&quot; | [[File:Rooted Node.jpg|20px]]
| &lt;math&gt;~&lt;/math&gt;
| &lt;math&gt;\mathrm{true}&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|-
| height=&quot;100px&quot; | [[File:Rooted Edge.jpg|20px]]
| &lt;math&gt;\texttt{(}~\texttt{)}&lt;/math&gt;
| &lt;math&gt;\mathrm{false}&lt;/math&gt;
| &lt;math&gt;0&lt;/math&gt;
|-
| height=&quot;100px&quot; | [[File:Cactus A Big.jpg|20px]]
| &lt;math&gt;a&lt;/math&gt;
| &lt;math&gt;a&lt;/math&gt;
| &lt;math&gt;a&lt;/math&gt;
|-
| height=&quot;120px&quot; | [[File:Cactus (A) Big.jpg|20px]]
| &lt;math&gt;\texttt{(} a \texttt{)}&lt;/math&gt;
| &lt;math&gt;\mathrm{not}~ a&lt;/math&gt;
| &lt;math&gt;\lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime&lt;/math&gt;
|-
| height=&quot;100px&quot; | [[File:Cactus ABC Big.jpg|50px]]
| &lt;math&gt;a ~ b ~ c&lt;/math&gt;
| &lt;math&gt;a ~\mathrm{and}~ b ~\mathrm{and}~ c&lt;/math&gt;
| &lt;math&gt;a \land b \land c&lt;/math&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus ((A)(B)(C)) Big.jpg|65px]]
| &lt;math&gt;\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}&lt;/math&gt;
| &lt;math&gt;a ~\mathrm{or}~ b ~\mathrm{or}~ c&lt;/math&gt;
| &lt;math&gt;a \lor b \lor c&lt;/math&gt;
|-
| height=&quot;120px&quot; | [[File:Cactus (A(B)) Big.jpg|60px]]
| &lt;math&gt;\texttt{(} a \texttt{(} b \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a ~\mathrm{implies}~ b
\\[6pt]
\mathrm{if}~ a ~\mathrm{then}~ b
\end{matrix}&lt;/math&gt;
| &lt;math&gt;a \Rightarrow b&lt;/math&gt;
|-
| height=&quot;120px&quot; | [[File:Cactus (A,B) Big.jpg|65px]]
| &lt;math&gt;\texttt{(} a \texttt{,} b \texttt{)}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a ~\mathrm{not~equal~to}~ b
\\[6pt]
a ~\mathrm{exclusive~or}~ b
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a \neq b
\\[6pt]
a + b
\end{matrix}&lt;/math&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus ((A,B)) Big.jpg|65px]]
| &lt;math&gt;\texttt{((} a \texttt{,} b \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a ~\mathrm{is~equal~to}~ b
\\[6pt]
a ~\mathrm{if~and~only~if}~ b
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a = b
\\[6pt]
a \Leftrightarrow b
\end{matrix}&lt;/math&gt;
|-
| height=&quot;120px&quot; | [[File:Cactus (A,B,C) Big.jpg|65px]]
| &lt;math&gt;\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~false}.
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
&amp; \bar{a} ~ b ~ c
\\
\lor &amp; a ~ \bar{b} ~ c
\\
\lor &amp; a ~ b ~ \bar{c}
\end{matrix}&lt;/math&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus ((A),(B),(C)) Big.jpg|65px]]
| &lt;math&gt;\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~true}.
\\[6pt]
\mathrm{partition~all}
\\
\mathrm{into}~ a, b, c.
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
&amp; a ~ \bar{b} ~ \bar{c}
\\
\lor &amp; \bar{a} ~ b ~ \bar{c}
\\
\lor &amp; \bar{a} ~ \bar{b} ~ c
\end{matrix}&lt;/math&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus (A,(B,C)) Big.jpg|90px]]
| &lt;math&gt;\texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
\mathrm{oddly~many~of}
\\
a, b, c
\\
\mathrm{are~true}.
\end{matrix}&lt;/math&gt;
|
&lt;p&gt;&lt;math&gt;a + b + c&lt;/math&gt;&lt;/p&gt;
&lt;br&gt;
&lt;p&gt;&lt;math&gt;\begin{matrix}
&amp; a ~ b ~ c
\\
\lor &amp; a ~ \bar{b} ~ \bar{c}
\\
\lor &amp; \bar{a} ~ b ~ \bar{c}
\\
\lor &amp; \bar{a} ~ \bar{b} ~ c
\end{matrix}&lt;/math&gt;&lt;/p&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus (X,(A),(B),(C)) Big.jpg|90px]]
| &lt;math&gt;\texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
\mathrm{partition}~ x
\\
\mathrm{into}~ a, b, c.
\\[6pt]
\mathrm{genus}~ x ~\mathrm{comprises}
\\
\mathrm{species}~ a, b, c.
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
&amp; \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c}
\\
\lor &amp; x ~ a ~ \bar{b} ~ \bar{c}
\\
\lor &amp; x ~ \bar{a} ~ b ~ \bar{c}
\\
\lor &amp; x ~ \bar{a} ~ \bar{b} ~ c
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Alt Text Work Area===

Table 1 is the first of several “Rosetta Stones” we'll use in this discussion to translate between different languages for the same subject matters.  The present Table displays equivalent expressions for frequently encountered propositional forms in four notations for propositional calculus.  The Table has four columns, labeled  “Graph”, “Expression”, “Interpretation”, and “Other Notations”, respectively.

• Column 1 “Graph” exhibits a logical graph for a commonly occurring propositional form.

• Column 2 “Expression” exhibits the text string transcription of the graph in Column 1.

• Column 3 “Interpretation” gives one or more verbal formulas for the graph in Column 1.

• Column 4 “Other Notations” shows several ways of notating the graph's logical meaning.

{|
| [[File:Rooted Node.jpg|20px]] || ~ || true || 1
|}

{|
| [[File:Rooted Edge.jpg|20px]] || ( ) || false || 0
|}

{|
| [[File:Cactus A Big.jpg|20px]] || a || a || a
|}

{|
| [[File:Cactus (A) Big.jpg|20px]] || ( a ) || not a || &amp;not;a &amp;#257; &amp;atilde; a′
|}

{|
| [[File:Cactus ABC Big.jpg|50px]] || abc || a and b and c || a &amp;and; b &amp;and; c
|}

{|
| [[File:Cactus ((A)(B)(C)) Big.jpg|65px]] || ((a)(b)(c)) || a or b or c || a &amp;or; b &amp;or; c
|}

{|
| [[File:Cactus (A(B)) Big.jpg|60px]]
| ( a ( b ))
|
{|
| a  implies  b
|-
| if  a  then  b
|}
| a &amp;rArr; b
|}

{|
| [[File:Cactus (A,B) Big.jpg|65px]]
| ( a , b )
|
{|
| a  not equal to  b
|-
| a  exclusive or  b
|}
|
{|
| a &amp;ne; b
|-
| a + b
|}
|}

{|
| [[File:Cactus ((A,B)) Big.jpg|65px]]
| (( a , b ))
|
{|
| a equal to b
|-
| a  if and only if  b
|}
|
{|
| a = b
|-
| a &amp;hArr; b
|}
|}

{|
| [[File:Cactus (A,B,C) Big.jpg|65px]]
| ( a , b , c )
| just one of a, b, c is false.
|
{|
| &amp;not;a &amp;and; b &amp;and; c
|-
| &amp;or; a &amp;and; &amp;not;b &amp;and; c
|-
| &amp;or; a &amp;and; b &amp;and; &amp;not;c
|}
|}

{|
| [[File:Cactus ((A),(B),(C)) Big.jpg|65px]]
| (( a ),( b ),( c ))
|
{|
| just one of a, b, c is true.
|-
| partition all into a, b, c.
|}
|
{|
|      a &amp;and; &amp;not;b &amp;and; &amp;not;c
|-
| &amp;or; &amp;not;a &amp;and; b &amp;and; &amp;not;c
|-
| &amp;or; &amp;not;a &amp;and; &amp;not;b &amp;and; c
|}
|}

{|
| [[File:Cactus (A,(B,C)) Big.jpg|90px]]
| ( a ,( b , c ))
| oddly many of a, b, c are true.
|
{|
| &amp;nbsp;&amp;nbsp; a + b + c
|-
{|
| &amp;nbsp;&amp;nbsp; a &amp;and; b &amp;and; c
|- 
| &amp;or; a &amp;and; &amp;not;b &amp;and; &amp;not;c
|-
| &amp;or; &amp;not;a &amp;and; b &amp;and; &amp;not;c
|-
| &amp;or; &amp;not;a &amp;and; &amp;not;b &amp;and; c
|}
|}

{|
| [[File:Cactus (X,(A),(B),(C)) Big.jpg|90px]]
| ( x ,( a ),( b ),( c ))
|
{|
| partition x into a, b, c.
|-
| genus x of species a, b, c.
|}
|
{|
| &amp;not;x &amp;and; &amp;not;a &amp;and; &amp;not;b &amp;and; &amp;not;c
|-
| &amp;or; x &amp;and; a &amp;and; &amp;not;b &amp;and; &amp;not;c
|-
| &amp;or; x &amp;and; &amp;not;a &amp;and; b &amp;and; &amp;not;c
|-
| &amp;or; x &amp;and; &amp;not;a &amp;and; &amp;not;b &amp;and; c
|}
|}

==Mathstodon Versions==

===Differential Logic and Dynamic Systems • Overview===
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview

❝Stand and unfold yourself.❞
— Hamlet • Francisco • 1.1.2

In modeling intelligent systems, whether we are trying to understand a natural system or engineer an artificial system, there has long been a tension or trade-off between dynamic paradigms and symbolic paradigms.  Dynamic models take their cue from physics, using quantitative measures and differential equations to model the evolution of a system’s state through time.  Symbolic models use logical methods to describe systems and their agents in qualitative terms, deriving logical consequences of a system’s description or an agent’s state of information.  Logic-based systems have tended to be static in character, largely because we have lacked a proper logical analogue of differential calculus.  The work laid out in this report is intended to address that lack.

This article develops a differential extension of propositional calculus and applies it to the analysis of dynamic systems whose states are described in qualitative logical terms.

The work pursued here is coordinated with a parallel application focusing on neural network systems but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.

===Differential Logic and Dynamic Systems • Review and Transition 1===
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Review_and_Transition

This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems.  It is useful to begin by summarizing essential material from previous reports.

Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable \(k\)-ary scope.

• A bracketed list of propositional expressions in the form \(\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}\) indicates that exactly one of the propositions \(e_1, e_2, \ldots, e_{k-1}, e_k\) is false.

• A concatenation of propositional expressions in the form \(e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k\) indicates that all of the propositions \(e_1, e_2, \ldots, e_{k-1}, e_k\) are true, in other words, that their logical conjunction is true.

All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms.  Strictly speaking, the concatenation form is dispensable in light of the bracketed form but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.

==Review and Transition (OEIS Version)==

This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems.&amp;nbsp; It is useful to begin by summarizing essential material from previous reports.

Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable &lt;math&gt;k&lt;/math&gt;-ary scope.

* A bracketed list of propositional expressions in the form &lt;math&gt;\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}&lt;/math&gt; indicates that exactly one of the propositions &lt;math&gt;e_1, e_2, \ldots, e_{k-1}, e_k&lt;/math&gt; is false.

* A concatenation of propositional expressions in the form &lt;math&gt;e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k&lt;/math&gt; indicates that all of the propositions &lt;math&gt;e_1, e_2, \ldots, e_{k-1}, e_k&lt;/math&gt; are true, in other words, that their [[logical conjunction]] is true.

All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms.&amp;nbsp; Strictly speaking, the concatenation form is dispensable in light of the bracketed form but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.

{| align=&quot;center&quot; style=&quot;font-size:larger; text-align:center; width:100%&quot;
| height=&quot;20px&quot; | &lt;math&gt;\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}&lt;/math&gt;
|-
| [[File:Syntax and Semantics of a Calculus for Propositional Logic 4.0.png|600px]]
|}

Table 1 is the first of several &amp;ldquo;Rosetta Stones&amp;rdquo; we'll use in this discussion to translate between different languages for the same subject matters.&amp;nbsp; In this case the Table displays equivalent expressions for simple examples of propositional forms in four notations for propositional calculus.

* Column 1 shows the logical graphs used to represent a number of simple propositional forms.
* Column 2 shows the traverse strings corresponding to the logical graphs in Column&amp;nbsp;1.
* Column 3 interprets the graph and string by means of conventional verbal formulas.
* Column 4 translates the interpretation into a number of symbolic notations.

This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob].  More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB].  Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.

While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives.  In contexts where parentheses are needed for other purposes &amp;ldquo;teletype&amp;rdquo; parentheses &lt;math&gt;\texttt{(} \ldots \texttt{)}&lt;/math&gt; or barred parentheses &lt;math&gt;(\!| \ldots |\!)&lt;/math&gt; may be used for logical operators.

The briefest expression for logical truth is the empty word, usually denoted by &lt;math&gt;{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}&lt;/math&gt; or &lt;math&gt;{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}&lt;/math&gt; in formal languages, where it forms the identity element for concatenation.  To make it visible in this text, it may be denoted by the equivalent expression &lt;math&gt;{}^{\backprime\backprime} \texttt{((} ~ \texttt{))} {}^{\prime\prime},&lt;/math&gt; or, especially if operating in an algebraic context, by a simple &lt;math&gt;{}^{\backprime\backprime} 1 {}^{\prime\prime}.&lt;/math&gt;  Also when working in an algebraic mode, the plus sign &lt;math&gt;{}^{\backprime\backprime} + {}^{\prime\prime}&lt;/math&gt; may be used for [[exclusive disjunction]].  For example, we have the following paraphrases of algebraic expressions by bracket expressions:

{| align=&quot;center&quot; cellpadding=&quot;6&quot; style=&quot;text-align:center&quot;
|
&lt;math&gt;\begin{matrix}
x + y ~=~ \texttt{(} x, y \texttt{)}
\\[6pt]
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}
\end{matrix}&lt;/math&gt;
|}

It is important to note that the last expressions are not equivalent to the triple bracket &lt;math&gt;\texttt{(} x, y, z \texttt{)}.&lt;/math&gt;

&lt;b&gt;Note.&lt;/b&gt; The usage that one often sees, of a plus sign &quot;&lt;math&gt;+&lt;/math&gt;&quot; to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts.  Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:

&lt;blockquote&gt;
The expression &lt;math&gt;x + y&lt;/math&gt; seems indeed uninterpretable, unless it be assumed that the things represented by &lt;math&gt;x&lt;/math&gt; and the things represented by &lt;math&gt;y&lt;/math&gt; are entirely separate;  that they embrace no individuals in common.  (Boole, 66).
&lt;/blockquote&gt;

It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189).  It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208).  Additional information, discussion, and references can be found in (Boole) and (Sty, 177&amp;ndash;263).  Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic.  For this reason, it will be avoided here.</text>
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      <text bytes="4453" sha1="5kk3ke7hodstv621ffuz50an6yt3pwp" xml:space="preserve">==Syntax and Semantics of a Calculus for Propositional Logic==

===PNG===

Table 1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives.

{| align=&quot;center&quot; style=&quot;font-size:large; text-align:center&quot;
|+ height=&quot;30px&quot; | &lt;math&gt;\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}&lt;/math&gt;
|-
| [[File:Syntax and Semantics of a Calculus for Propositional Logic 4.0.png|600px]]
|}

===Wiki + LaTeX + JPG===

Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable &lt;math&gt;k&lt;/math&gt;-ary scope.

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; |
&lt;math&gt;\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Graph}&lt;/math&gt;
| &lt;math&gt;\text{Expression}&lt;/math&gt;
| &lt;math&gt;\text{Interpretation}&lt;/math&gt;
| &lt;math&gt;\text{Other Notations}&lt;/math&gt;
|-
| height=&quot;100px&quot; | [[File:Rooted Node.jpg|20px]]
| &lt;math&gt;~&lt;/math&gt;
| &lt;math&gt;\mathrm{true}&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|-
| height=&quot;100px&quot; | [[File:Rooted Edge.jpg|20px]]
| &lt;math&gt;\texttt{(}~\texttt{)}&lt;/math&gt;
| &lt;math&gt;\mathrm{false}&lt;/math&gt;
| &lt;math&gt;0&lt;/math&gt;
|-
| height=&quot;100px&quot; | [[File:Cactus A Big.jpg|20px]]
| &lt;math&gt;a&lt;/math&gt;
| &lt;math&gt;a&lt;/math&gt;
| &lt;math&gt;a&lt;/math&gt;
|-
| height=&quot;120px&quot; | [[File:Cactus (A) Big.jpg|20px]]
| &lt;math&gt;\texttt{(} a \texttt{)}&lt;/math&gt;
| &lt;math&gt;\mathrm{not}~ a&lt;/math&gt;
| &lt;math&gt;\lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime&lt;/math&gt;
|-
| height=&quot;100px&quot; | [[File:Cactus ABC Big.jpg|50px]]
| &lt;math&gt;a ~ b ~ c&lt;/math&gt;
| &lt;math&gt;a ~\mathrm{and}~ b ~\mathrm{and}~ c&lt;/math&gt;
| &lt;math&gt;a \land b \land c&lt;/math&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus ((A)(B)(C)) Big.jpg|65px]]
| &lt;math&gt;\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}&lt;/math&gt;
| &lt;math&gt;a ~\mathrm{or}~ b ~\mathrm{or}~ c&lt;/math&gt;
| &lt;math&gt;a \lor b \lor c&lt;/math&gt;
|-
| height=&quot;120px&quot; | [[File:Cactus (A(B)) Big.jpg|60px]]
| &lt;math&gt;\texttt{(} a \texttt{(} b \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a ~\mathrm{implies}~ b
\\[6pt]
\mathrm{if}~ a ~\mathrm{then}~ b
\end{matrix}&lt;/math&gt;
| &lt;math&gt;a \Rightarrow b&lt;/math&gt;
|-
| height=&quot;120px&quot; | [[File:Cactus (A,B) Big.jpg|65px]]
| &lt;math&gt;\texttt{(} a \texttt{,} b \texttt{)}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a ~\mathrm{not~equal~to}~ b
\\[6pt]
a ~\mathrm{exclusive~or}~ b
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a \neq b
\\[6pt]
a + b
\end{matrix}&lt;/math&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus ((A,B)) Big.jpg|65px]]
| &lt;math&gt;\texttt{((} a \texttt{,} b \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a ~\mathrm{is~equal~to}~ b
\\[6pt]
a ~\mathrm{if~and~only~if}~ b
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a = b
\\[6pt]
a \Leftrightarrow b
\end{matrix}&lt;/math&gt;
|-
| height=&quot;120px&quot; | [[File:Cactus (A,B,C) Big.jpg|65px]]
| &lt;math&gt;\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~false}.
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
&amp; \bar{a} ~ b ~ c
\\
\lor &amp; a ~ \bar{b} ~ c
\\
\lor &amp; a ~ b ~ \bar{c}
\end{matrix}&lt;/math&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus ((A),(B),(C)) Big.jpg|65px]]
| &lt;math&gt;\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~true}.
\\[6pt]
\mathrm{partition~all}
\\
\mathrm{into}~ a, b, c.
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
&amp; a ~ \bar{b} ~ \bar{c}
\\
\lor &amp; \bar{a} ~ b ~ \bar{c}
\\
\lor &amp; \bar{a} ~ \bar{b} ~ c
\end{matrix}&lt;/math&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus (A,(B,C)) Big.jpg|90px]]
| &lt;math&gt;\texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
\mathrm{oddly~many~of}
\\
a, b, c
\\
\mathrm{are~true}.
\end{matrix}&lt;/math&gt;
|
&lt;p&gt;&lt;math&gt;a + b + c&lt;/math&gt;&lt;/p&gt;
&lt;br&gt;
&lt;p&gt;&lt;math&gt;\begin{matrix}
&amp; a ~ b ~ c
\\
\lor &amp; a ~ \bar{b} ~ \bar{c}
\\
\lor &amp; \bar{a} ~ b ~ \bar{c}
\\
\lor &amp; \bar{a} ~ \bar{b} ~ c
\end{matrix}&lt;/math&gt;&lt;/p&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus (X,(A),(B),(C)) Big.jpg|90px]]
| &lt;math&gt;\texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
\mathrm{partition}~ x
\\
\mathrm{into}~ a, b, c.
\\[6pt]
\mathrm{genus}~ x ~\mathrm{comprises}
\\
\mathrm{species}~ a, b, c.
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
&amp; \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c}
\\
\lor &amp; x ~ a ~ \bar{b} ~ \bar{c}
\\
\lor &amp; x ~ \bar{a} ~ b ~ \bar{c}
\\
\lor &amp; x ~ \bar{a} ~ \bar{b} ~ c
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;</text>
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      <text bytes="4453" sha1="5kk3ke7hodstv621ffuz50an6yt3pwp" xml:space="preserve">==Syntax and Semantics of a Calculus for Propositional Logic==

===PNG===

Table 1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives.

{| align=&quot;center&quot; style=&quot;font-size:large; text-align:center&quot;
|+ height=&quot;30px&quot; | &lt;math&gt;\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}&lt;/math&gt;
|-
| [[File:Syntax and Semantics of a Calculus for Propositional Logic 4.0.png|600px]]
|}

===Wiki + LaTeX + JPG===

Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable &lt;math&gt;k&lt;/math&gt;-ary scope.

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; |
&lt;math&gt;\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Graph}&lt;/math&gt;
| &lt;math&gt;\text{Expression}&lt;/math&gt;
| &lt;math&gt;\text{Interpretation}&lt;/math&gt;
| &lt;math&gt;\text{Other Notations}&lt;/math&gt;
|-
| height=&quot;100px&quot; | [[File:Rooted Node.jpg|20px]]
| &lt;math&gt;~&lt;/math&gt;
| &lt;math&gt;\mathrm{true}&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|-
| height=&quot;100px&quot; | [[File:Rooted Edge.jpg|20px]]
| &lt;math&gt;\texttt{(}~\texttt{)}&lt;/math&gt;
| &lt;math&gt;\mathrm{false}&lt;/math&gt;
| &lt;math&gt;0&lt;/math&gt;
|-
| height=&quot;100px&quot; | [[File:Cactus A Big.jpg|20px]]
| &lt;math&gt;a&lt;/math&gt;
| &lt;math&gt;a&lt;/math&gt;
| &lt;math&gt;a&lt;/math&gt;
|-
| height=&quot;120px&quot; | [[File:Cactus (A) Big.jpg|20px]]
| &lt;math&gt;\texttt{(} a \texttt{)}&lt;/math&gt;
| &lt;math&gt;\mathrm{not}~ a&lt;/math&gt;
| &lt;math&gt;\lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime&lt;/math&gt;
|-
| height=&quot;100px&quot; | [[File:Cactus ABC Big.jpg|50px]]
| &lt;math&gt;a ~ b ~ c&lt;/math&gt;
| &lt;math&gt;a ~\mathrm{and}~ b ~\mathrm{and}~ c&lt;/math&gt;
| &lt;math&gt;a \land b \land c&lt;/math&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus ((A)(B)(C)) Big.jpg|65px]]
| &lt;math&gt;\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}&lt;/math&gt;
| &lt;math&gt;a ~\mathrm{or}~ b ~\mathrm{or}~ c&lt;/math&gt;
| &lt;math&gt;a \lor b \lor c&lt;/math&gt;
|-
| height=&quot;120px&quot; | [[File:Cactus (A(B)) Big.jpg|60px]]
| &lt;math&gt;\texttt{(} a \texttt{(} b \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a ~\mathrm{implies}~ b
\\[6pt]
\mathrm{if}~ a ~\mathrm{then}~ b
\end{matrix}&lt;/math&gt;
| &lt;math&gt;a \Rightarrow b&lt;/math&gt;
|-
| height=&quot;120px&quot; | [[File:Cactus (A,B) Big.jpg|65px]]
| &lt;math&gt;\texttt{(} a \texttt{,} b \texttt{)}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a ~\mathrm{not~equal~to}~ b
\\[6pt]
a ~\mathrm{exclusive~or}~ b
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a \neq b
\\[6pt]
a + b
\end{matrix}&lt;/math&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus ((A,B)) Big.jpg|65px]]
| &lt;math&gt;\texttt{((} a \texttt{,} b \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a ~\mathrm{is~equal~to}~ b
\\[6pt]
a ~\mathrm{if~and~only~if}~ b
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
a = b
\\[6pt]
a \Leftrightarrow b
\end{matrix}&lt;/math&gt;
|-
| height=&quot;120px&quot; | [[File:Cactus (A,B,C) Big.jpg|65px]]
| &lt;math&gt;\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~false}.
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
&amp; \bar{a} ~ b ~ c
\\
\lor &amp; a ~ \bar{b} ~ c
\\
\lor &amp; a ~ b ~ \bar{c}
\end{matrix}&lt;/math&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus ((A),(B),(C)) Big.jpg|65px]]
| &lt;math&gt;\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~true}.
\\[6pt]
\mathrm{partition~all}
\\
\mathrm{into}~ a, b, c.
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
&amp; a ~ \bar{b} ~ \bar{c}
\\
\lor &amp; \bar{a} ~ b ~ \bar{c}
\\
\lor &amp; \bar{a} ~ \bar{b} ~ c
\end{matrix}&lt;/math&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus (A,(B,C)) Big.jpg|90px]]
| &lt;math&gt;\texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
\mathrm{oddly~many~of}
\\
a, b, c
\\
\mathrm{are~true}.
\end{matrix}&lt;/math&gt;
|
&lt;p&gt;&lt;math&gt;a + b + c&lt;/math&gt;&lt;/p&gt;
&lt;br&gt;
&lt;p&gt;&lt;math&gt;\begin{matrix}
&amp; a ~ b ~ c
\\
\lor &amp; a ~ \bar{b} ~ \bar{c}
\\
\lor &amp; \bar{a} ~ b ~ \bar{c}
\\
\lor &amp; \bar{a} ~ \bar{b} ~ c
\end{matrix}&lt;/math&gt;&lt;/p&gt;
|-
| height=&quot;160px&quot; | [[File:Cactus (X,(A),(B),(C)) Big.jpg|90px]]
| &lt;math&gt;\texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
\mathrm{partition}~ x
\\
\mathrm{into}~ a, b, c.
\\[6pt]
\mathrm{genus}~ x ~\mathrm{comprises}
\\
\mathrm{species}~ a, b, c.
\end{matrix}&lt;/math&gt;
|
&lt;math&gt;\begin{matrix}
&amp; \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c}
\\
\lor &amp; x ~ a ~ \bar{b} ~ \bar{c}
\\
\lor &amp; x ~ \bar{a} ~ b ~ \bar{c}
\\
\lor &amp; x ~ \bar{a} ~ \bar{b} ~ c
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;</text>
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  <page>
    <title>Talk:Motivation and emotion/Book/2026/Cognitive versus affective empathy</title>
    <ns>1</ns>
    <id>331738</id>
    <revision>
      <id>2829886</id>
      <timestamp>2026-08-31T11:09:23Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>Created page with &quot;==Focus questions== Great progress. The titles and layout look very clear and well considered. The second and third questions don't have capitals at the beginning.  --~~~~&quot;</comment>
      <origin>2829886</origin>
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      <text bytes="302" sha1="jpuc4c983ooclov5p1txl2nzn0tkohf" xml:space="preserve">==Focus questions==
Great progress. The titles and layout look very clear and well considered. The second and third questions don't have capitals at the beginning. 
--[[User:U3228742|U3228742]] ([[User talk:U3228742|discuss]] • [[Special:Contributions/U3228742|contribs]]) 11:08, 31 August 2026 (UTC)</text>
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  <page>
    <title>User:JonAwbrey/Figures and Tables 50</title>
    <ns>2</ns>
    <id>331739</id>
    <revision>
      <id>2829891</id>
      <timestamp>2026-08-31T11:14:04Z</timestamp>
      <contributor>
        <username>JonAwbrey</username>
        <id>29537</id>
      </contributor>
      <comment>JonAwbrey moved page [[User:JonAwbrey/Figures and Tables 50]] to [[User:JonAwbrey/Figures and Tables 54]]</comment>
      <origin>2829891</origin>
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      <text bytes="50" sha1="6nag8qgtkqotqnt8ook490fwng2t6hd" xml:space="preserve">#REDIRECT [[User:JonAwbrey/Figures and Tables 54]]</text>
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    <revision>
      <id>2829899</id>
      <parentid>2829891</parentid>
      <timestamp>2026-08-31T11:54:51Z</timestamp>
      <contributor>
        <username>JonAwbrey</username>
        <id>29537</id>
      </contributor>
      <comment>delete redirect</comment>
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